mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case intro.refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
ι : Type u_4
inst✝¹ : Finite ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
val✝ : Fintype ι
this : ∀ (n : ℕ), MeasurableSet (⋂ i, spanningSets (μ i) n)
⊢ ⋃ i, ⋂ i_1, spanningSets (μ i_1) i = univ

Tactic:
rw [iUnion_iInter_of_monotone]

Goal state after:
case intro.refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
ι : Type u_4
inst✝¹ : Finite ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
val✝ : Fintype ι
this : ∀ (n : ℕ), MeasurableSet (⋂ i, spanningSets (μ i) n)
⊢ ⋂ i, ⋃ j, spanningSets (μ i) j = univ

case intro.refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
ι : Type u_4
inst✝¹ : Finite ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
val✝ : Fintype ι
this : ∀ (n : ℕ), MeasurableSet (⋂ i, spanningSets (μ i) n)
⊢ ∀ (i : ι), Monotone (spanningSets (μ i))
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
hs : s ∈ memPartition t n
⊢ s \ t n = s ∩ t n ∨ s \ t n = s \ t n

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
hs : s ∈ memPartition t n
⊢ s ∩ t n = s ∩ t n ∨ s ∩ t n = s \ t n

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s u : Set α
hu : u ∈ memPartition t n
ht✝ : MeasurableSet u
⊢ u = ⋃₀ ↑{u}

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
⊢ ↑∅ ⊆ memPartition t n

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
⊢ ∅ = ⋃₀ ↑∅

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
hf : MeasurableEmbedding f
inst✝ : SigmaFinite μ
⊢ ∀ (i : ℕ), f '' spanningSets μ i ∪ (range f)ᶜ ∈ univ

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_1
α : Type u_1
t : ℕ → Set α
u : Set α
hu : ∃ i, u ∈ memPartition t i
⊢ MeasurableSet u

Tactic:
obtain ⟨n, hun⟩ := hu

Goal state after:
case refine_1
α : Type u_1
t : ℕ → Set α
u : Set α
n : ℕ
hun : u ∈ memPartition t n
⊢ MeasurableSet u
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
inst✝ : IsEmpty ι
μ : ι → Measure α
⊢ (sum μ) univ = 0

Tactic:
sum_apply _ MeasurableSet.univ,

Goal state after:
α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
inst✝ : IsEmpty ι
μ : ι → Measure α
⊢ ∑' (i : ι), (μ i) univ = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0✝ : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
m0 : MeasurableSpace α
μ : Measure α
⊢ μ ∅ < ∞

Tactic:
by simp

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

Goal state before:
pp: case e_6
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ ⋃ i ∈ s, Iic i = univ

Tactic:
simp only [← bex_def] at hst

Goal state after:
case e_6
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
hst : ∀ (x : α), ∃ x_1, ∃ (_ : x_1 ∈ s), x ≤ x_1
⊢ ⋃ i ∈ s, Iic i = univ
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ M • (μ.restrict (⋃ C ∈ Cs, P C)) t ≤ (M • μ.restrict (⋃ i, s i)) t

Tactic:
by
      rw [Measure.smul_apply]
      exact nsmul_le_nsmul_right (μ.restrict_mono_set iUnion_P t) M

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s : Set α
⊢ 1 ≠ ∞

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : SecondCountableTopology α
inst✝¹ : DenselyOrdered α
inst✝ : NoMinOrder α
s : Set α
hd : Dense s
⊢ ∀ (x : α), IsBot x → x ∈ s

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hβ : SecondCountableTopology β
⊢ MeasurableSet s

Tactic:
· let F : Set β → Set α := fun a ↦ {y | ∃ b, IsOpen b ∧ y ∈ b ∧ b ×ˢ a ⊆ s}
    have A : ∀ a, IsOpen (F a) := by
      intro a
      apply isOpen_iff_forall_mem_open.2
      rintro y ⟨b, b_open, yb, hb⟩
      exact ⟨b, fun z zb ↦ ⟨b, b_open, zb, hb⟩, b_open, yb⟩
    have : s = ⋃ a ∈ countableBasis β, F a ×ˢ a := by
      apply Subset.antisymm
      · rintro ⟨y1, y2⟩ hy
        rcases isOpen_prod_iff.1 hs y1 y2 hy with ⟨u, v, u_open, v_open, yu, yv, huv⟩
        obtain ⟨a, ha, ya, au⟩ : ∃ a ∈ countableBasis β, y2 ∈ a ∧ a ⊆ v :=
          IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis β) yv v_open
        simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop]
        exact ⟨a, ⟨u, u_open, yu, (Set.prod_mono_right au).trans huv⟩, ha, ya⟩
      · rintro ⟨y1, y2⟩ hy
        simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop] at hy
        rcases hy with ⟨a, ⟨b, -, yb, hb⟩, -, ya⟩
        exact hb (mem_prod.2 ⟨yb, ya⟩)
    rw [this]
    apply MeasurableSet.biUnion (countable_countableBasis β) (fun a ha ↦ ?_)
    exact (A a).measurableSet.prod (isOpen_of_mem_countableBasis ha).measurableSet

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α : Type u_1
γ : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSingletonClass α
f : α → γ
s : Set α
hf : ContinuousOn f s
hs : sᶜ.Countable
⊢ Measurable (s.domRestrict f)

Tactic:
exact (continuousOn_iff_continuous_domRestrict.1 hf).measurable

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
⊢ hausdorffEDist s (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) +
      hausdorffEDist (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) t =
    hausdorffEDist (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) t

Tactic:
by simp

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

Goal state before:
pp: X : Type u
inst✝ : PseudoMetricSpace X
x : X
E : Set X
hx : x ∈ E
δ : ℝ
⊢ ball x δ ⊆ thickening δ {x}

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
x : α
hx : x < a
z : α
hzt : z ∈ t
hzx : z ≤ x
⊢ ∃ i, ∃ (_ : i ∈ t) (_ : i < a), i ≤ x

Tactic:
exact ⟨z, hzt, hzx.trans_lt hx, hzx⟩

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l u, l < u ∧ Ico l u = S}

Tactic:
simpa only [exists_prop, mem_univ, true_and] using
    (@dense_univ α _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>
      mem_univ _

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : SecondCountableTopology α
inst✝¹ : DenselyOrdered α
inst✝ : NoMaxOrder α
s : Set α
hd : Dense s
⊢ ∀ (x : α), IsTop x → x ∈ s

Tactic:
by simp

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

Goal state before:
pp: β : Type u_2
α : Type u_5
inst✝ : MeasurableSpace α
b : β
⊢ ↑(const α b).range ⊆ ↑{b}

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
r : α → β → Prop
f : α →ₛ β
h : ∀ (b : β), MeasurableSet {a | r a b}
⊢ MeasurableSet {a | r a (f a)}

Tactic:
by
  have : { a | r a (f a) } = ⋃ b ∈ range f, { a | r a b } ∩ f ⁻¹' {b} := by
    ext a
    suffices r a (f a) ↔ ∃ i, r a (f i) ∧ f a = f i by simpa
    exact ⟨fun h => ⟨a, ⟨h, rfl⟩⟩, fun ⟨a', ⟨h', e⟩⟩ => e.symm ▸ h'⟩
  rw [this]
  exact
    MeasurableSet.biUnion f.finite_range.countable fun b _ =>
      MeasurableSet.inter (h b) (f.measurableSet_fiber _)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
s : Set α
hs : MeasurableSet sᶜ
f g : α →ₛ β
⊢ ⇑(piecewise sᶜ hs f g) = ⇑(piecewise s ⋯ g f)

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
f g : α →ₛ β
⊢ ⇑(piecewise univ ⋯ f g) = ⇑f

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
f g : α →ₛ β
⊢ ⇑(piecewise ∅ ⋯ f g) = ⇑g

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Nonempty α
inst✝ : One β
x : β
⊢ x ∈ SimpleFunc.range 1 ↔ x ∈ {1}

Tactic:
by simp

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

Goal state before:
pp: α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
⊢ ∅ ∈ 𝓝ˢ ∅

Tactic:
by simp

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

Goal state before:
pp: α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
⊢ thickening 1 (f '' ∅) ⊆ U

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: case a.refine_1
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hC : IsCountablySpanning C
s : Set α
hs : s ∈ C
t : ℕ → Set β
h1t : ∀ (n : ℕ), t n ∈ D
h2t : ⋃ n, t n = univ
n : ℕ
⊢ MeasurableSet (s ×ˢ t n)

Tactic:
apply measurableSet_generateFrom

Goal state after:
case a.refine_1
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hC : IsCountablySpanning C
s : Set α
hs : s ∈ C
t : ℕ → Set β
h1t : ∀ (n : ℕ), t n ∈ D
h2t : ⋃ n, t n = univ
n : ℕ
⊢ s ×ˢ t n ∈ image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: case a.refine_2
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hD : IsCountablySpanning D
s : Set β
hs : s ∈ D
t : ℕ → Set α
h1t : ∀ (n : ℕ), t n ∈ C
h2t : ⋃ n, t n = univ
n : ℕ
⊢ MeasurableSet (t n ×ˢ s)

Tactic:
apply measurableSet_generateFrom

Goal state after:
case a.refine_2
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hD : IsCountablySpanning D
s : Set β
hs : s ∈ D
t : ℕ → Set α
h1t : ∀ (n : ℕ), t n ∈ C
h2t : ⋃ n, t n = univ
n : ℕ
⊢ t n ×ˢ s ∈ image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : CompleteLattice β
inst✝³ : OrderClosedTopology β
inst✝² : Zero β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
a : α
hf : Measurable f
h_zero : 0 = ⊥
n : ℕ
⊢ (approx i f n) a ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k

Tactic:
rw [approx_apply a hf, h_zero]
    refine Finset.sup_le fun k _ => ?_
    split_ifs with h
    · exact le_iSup_of_le k (le_iSup (fun _ : i k ≤ f a => i k) h)
    · exact bot_le

Goal state after:
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: q : ℚ
⊢ ↑q + 1 ≠ ↑q + 2

Tactic:
by simp

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

Goal state before:
pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
hm : f.FinMeasSupp μ
hf : ∀ᵐ (a : α) ∂μ, f a ≠ ∞
a : ℝ≥0∞
ha✝ : a ∈ f.range
ha : a ≠ ∞
⊢ a * μ (⇑f ⁻¹' {a}) < ∞

Tactic:
by_cases ha0 : a = 0
    · subst a
      simp
    · exact mul_lt_top ha.lt_top (finMeasSupp_iff.1 hm _ ha0)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: a : ℝ
this : IsLUB (range Rat.cast ∩ Iio a) a
b : ℚ
x✝ : b ∈ Rat.cast ⁻¹' Iio a
⊢ Iio ↑b ∈ ⋃ a, {Iio ↑a}

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: a : ℝ
this : IsGLB (range Rat.cast ∩ Ioi a) a
b : ℚ
x✝ : b ∈ Rat.cast ⁻¹' Ioi a
⊢ Ioi ↑b ∈ ⋃ a, {Ioi ↑a}

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
n : ℕ
⊢ Ioo (-(↑n + 1)) (↑n + 1) = Ioo ↑(-↑(n + 1)) ↑(↑n + 1)

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
⊢ Measurable fun r ↦ (↑r, ∞).1 * (↑r, ∞).2

Tactic:
simp only [ENNReal.mul_top', ENNReal.coe_eq_zero]
    exact measurable_const.piecewise (measurableSet_singleton _) measurable_const

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
⊢ Measurable fun r ↦ (∞, ↑r).1 - (∞, ↑r).2

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
⊢ Measurable fun r ↦ (↑r, ∞).1 - (↑r, ∞).2

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
x✝ : ℝ≥0
⊢ Measurable fun x ↦ x✝ • x

Tactic:
ENNReal.smul_def

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
x✝ : ℝ≥0
⊢ Measurable fun x ↦ ↑x✝ • x
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
x✝ : ℝ≥0∞
⊢ Measurable fun x ↦ x • x✝

Tactic:
ENNReal.smul_def

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
mα : MeasurableSpace α
x✝ : ℝ≥0∞
⊢ Measurable fun x ↦ ↑x • x✝
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
f : EReal → α
h : Measurable fun p ↦ f ↑p
⊢ {⊥, ⊤}.Finite

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
F0 : Subtype p → δ → α := fun j x ↦ ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x ↦ F0 (reparam x j) x
hF1 : F1 = fun x ↦ F0 (reparam x j) x
g : ℕ → Subtype p := Classical.choose ⋯
Z : ∀ (x : δ), ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k
x : δ
A : reparam x j = if x ∈ m j then j else g (Nat.find ⋯)
hjx : x ∈ m j
⊢ F1 x = F0 j x

Tactic:
have : reparam x j = j := by rw [A, if_pos hjx]

Goal state after:
case pos
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝¹ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this✝ : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
F0 : Subtype p → δ → α := fun j x ↦ ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x ↦ F0 (reparam x j) x
hF1 : F1 = fun x ↦ F0 (reparam x j) x
g : ℕ → Subtype p := Classical.choose ⋯
Z : ∀ (x : δ), ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k
x : δ
A : reparam x j = if x ∈ m j then j else g (Nat.find ⋯)
hjx : x ∈ m j
this : reparam x j = j
⊢ F1 x = F0 j x
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : PseudoEMetricSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : MeasurableSpace β
f✝ : β → α
inst✝² : Zero α
inst✝¹ : Preorder α
f : β → α
hf : 0 ≤ f
hfm : Measurable f
inst✝ : SeparableSpace ↑(range f ∪ {0})
n : ℕ
⊢ 0 ∈ range f ∪ {0}

Tactic:
by simp

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
⊢ MeasurableSet ∅ ∧ ∅ ⊆ ∅ ∧ ∀ x ∈ ∅, dist (f x) (0 x) < ε

Tactic:
by simp

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
M : ∀ (K : Set (X × Y)), IsCompact K → ∃ g s, MeasurableSet s ∧ K ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
⊢ ∃ g, ∀ (x : X × Y), dist (f x) (g x) < ε

Tactic:
obtain ⟨g, s, s_meas, fs, hg⟩ : ∃ (g : SimpleFunc (X × Y) α) (s : Set (X × Y)),
    MeasurableSet s ∧ tsupport f ⊆ s ∧ ∀ (x : X × Y), x ∈ s → dist (f x) (g x) < ε := M _ h'f

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
M : ∀ (K : Set (X × Y)), IsCompact K → ∃ g s, MeasurableSet s ∧ K ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
g : SimpleFunc (X × Y) α
s : Set (X × Y)
s_meas : MeasurableSet s
fs : tsupport f ⊆ s
hg : ∀ x ∈ s, dist (f x) (g x) < ε
⊢ ∃ g, ∀ (x : X × Y), dist (f x) (g x) < ε
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ ∫⁻ (x : α), 0 ∂μ = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
⊢ (⨅ x, f x) * μ univ = ∫⁻ (y : α), ⨅ x, f x ∂μ

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
⊢ ∫⁻ (y : α), ⨆ x, f x ∂μ = (⨆ x, f x) * μ univ

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ (⨅ x ∈ s, f x) * μ s = ∫⁻ (y : α) in s, ⨅ x ∈ s, f x ∂μ

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (y : α) in s, ⨆ x ∈ s, f x ∂μ = (⨆ x ∈ s, f x) * μ s

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
G : Type u_6
inst✝⁴ : Group G
inst✝³ : MulAction G β
inst✝² : TopologicalSpace G
inst✝¹ : ContinuousInv G
inst✝ : ContinuousSMul G β
c : α → G
hc : AEStronglyMeasurable c μ
h : AEStronglyMeasurable (fun x ↦ c x • f x) μ
⊢ (fun i ↦ (c i)⁻¹ • c i • f i) =ᵐ[μ] f

Tactic:
by simp

Goal state after: