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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ u < μ s + μ t
⊢ (s ∩ t).Nonempty

Tactic:
rw [← Set.not_disjoint_iff_nonempty_inter]
  contrapose! h
  calc
    μ s + μ t = μ (s ∪ t) := (measure_union h ht).symm
    _ ≤ μ u := measure_mono (union_subset h's h't)

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)

Tactic:
rcases Countable.exists_injective_nat ι with ⟨e, he⟩

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
e : ι → ℕ
he : Injective e
⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
inst✝ : IsFiniteMeasure (map f μ)
⊢ μ univ < ∞

Tactic:
by
    rw [← Set.preimage_univ (f := f), ← map_apply_of_aemeasurable hf .univ]
    exact IsFiniteMeasure.measure_univ_lt_top

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
⊢ measureUnivNNReal μ = 0 ↔ ↑(measureUnivNNReal μ) = 0

Tactic:
norm_cast

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (s i) μ
hd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s
k : ι
hk : μ (s k) ≠ ∞
this : ∀ t ⊆ s k, μ t ≠ ∞
⊢ μ (⋂ i, s i) = μ (s k) - ⨆ i, μ (s k) - μ (s i)

Tactic:
←
    ENNReal.sub_sub_cancel hk (measure_mono (iInter_subset _ k)),

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (s i) μ
hd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s
k : ι
hk : μ (s k) ≠ ∞
this : ∀ t ⊆ s k, μ t ≠ ∞
⊢ μ (s k) - (μ (s k) - μ (⋂ i, s i)) = μ (s k) - ⨆ i, μ (s k) - μ (s i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
p : α → Prop
hp : NullMeasurableSet {a | p a} μ
⊢ (∀ᵐ (a : α) ∂μ, p a) ↔ μ {a | p a} = μ univ

Tactic:
by
  rw [← ae_eq_univ_iff_measure_eq hp, eventuallyEq_univ, eventually_iff]

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

Goal state before:
pp: α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
⊢ {i | 0 < (sum (sfiniteSeq μ)) (As i)} ⊆ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}

Tactic:
intro i

Goal state after:
α : Type u_1
ι : Type u_4
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
As : ι → Set α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) (sum (sfiniteSeq μ))
As_disj : Pairwise (AEDisjoint (sum (sfiniteSeq μ)) on As)
i : ι
⊢ i ∈ {i | 0 < (sum (sfiniteSeq μ)) (As i)} → i ∈ ⋃ n, {i | 0 < (sfiniteSeq μ n) (As i)}
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ μ (⋂ i, s i) = ⨅ i, μ (s i)

Tactic:
iInf_congr <| fun i ↦ measure_congr <| hst i,

Goal state after:
α : Type u_1
ι : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : Preorder ι
inst✝¹ : IsCodirectedOrder ι
inst✝ : atBot.IsCountablyGenerated
s : ι → Set α
hs : ∀ᵐ (ω : α) ∂μ, Monotone fun x ↦ ω ∈ s x
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
i : ι
hi : μ (s i) ≠ ∞
this : Nonempty ι
t : ι → Set α := fun i ↦ s i ∩ {ω | Monotone fun x ↦ ω ∈ s x}
hst : ∀ (i : ι), s i =ᵐ[μ] t i
hMono : Monotone t
⊢ μ (⋂ i, s i) = ⨅ i, μ (t i)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ✝ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Finite ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), IsFiniteMeasure (μ i)
⊢ (sum μ) univ < ∞

Tactic:
cases nonempty_fintype ι
    simp [measure_lt_top]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : IsEmpty α
⊢ ⋃ i, ∅ = univ

Tactic:
simp [eq_iff_true_of_subsingleton]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t✝ : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t✝ ⊆ u
_hu : MeasurableSet u
s t : Set α
hst : s ⊆ t
⊢ ⨅ μ ∈ toOuterMeasure '' m, μ s ≤ μ t

Tactic:
iInf_image

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
m : Set (Measure α)
s✝ : Set α
hs : MeasurableSet s✝
t✝ : Set α
μ : Measure α
hμ : μ ∈ m
u : Set α
htu : t✝ ⊆ u
_hu : MeasurableSet u
s t : Set α
hst : s ⊆ t
⊢ ⨅ b ∈ m, b.toOuterMeasure s ≤ μ t
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
w : ℕ → Set α := fun n ↦ toMeasurable μ (t ∩ v n)
hw : ∀ (n : ℕ), μ (w n) < ∞
t' : Set α := ⋃ n, toMeasurable μ (t ∩ disjointed w n)
ht' : t' = ⋃ n, toMeasurable μ (t ∩ disjointed w n)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
⊢ ∀ (i : ℕ), MeasurableSet (disjointed w i)

Tactic:
intro i

Goal state after:
case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞
w : ℕ → Set α := fun n ↦ toMeasurable μ (t ∩ v n)
hw : ∀ (n : ℕ), μ (w n) < ∞
t' : Set α := ⋃ n, toMeasurable μ (t ∩ disjointed w n)
ht' : t' = ⋃ n, toMeasurable μ (t ∩ disjointed w n)
tt' : t ⊆ t'
u : Set α
hu : MeasurableSet u
i : ℕ
⊢ MeasurableSet (disjointed w i)
Module: Mathlib.Order.Filter.CountableSeparatingOn

Goal state before:
pp: case refine_2
α : Type u_1
l : Filter α
inst✝ : CountableInterFilter l
p : Set α → Prop
s✝ : Set α
h : HasCountableSeparatingOn α p s✝
hs : s✝ ∈ l
hl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l
S : Set (Set α)
hSc : S.Countable
hSp : ∀ s ∈ S, p s
hS : ∀ x ∈ s✝, ∀ y ∈ s✝, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
x y : α
hx : (x ∈ s✝ ∧ ∀ (t : Set α), t ∈ S ∧ t ∈ l.sets → x ∈ t) ∧ ∀ i ∈ S, iᶜ ∈ l → x ∉ i
hy : (y ∈ s✝ ∧ ∀ (t : Set α), t ∈ S ∧ t ∈ l.sets → y ∈ t) ∧ ∀ i ∈ S, iᶜ ∈ l → y ∉ i
s : Set α
hsS : s ∈ S
⊢ x ∈ s ↔ y ∈ s

Tactic:
cases hl s (hSp s hsS) with
    | inl hsl => simp only [hx.1.2 s ⟨hsS, hsl⟩, hy.1.2 s ⟨hsS, hsl⟩]
    | inr hsl => simp only [hx.2 s hsS hsl, hy.2 s hsS hsl]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
⊢ ∀ (i : ℕ), ν₁.restrict (spanningSets μ i) = ν₂.restrict (spanningSets μ i)

Tactic:
intro i

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
inst✝ : SigmaFinite μ
h : μ + ν₁ = μ + ν₂
i : ℕ
⊢ ν₁.restrict (spanningSets μ i) = ν₂.restrict (spanningSets μ i)
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᵐ[μ.restrict s] f

Tactic:
ae_restrict_eq hs

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 s] f
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
⊢ ∀ (i : ℕ), (c • μ) (spanningSets μ i) < ∞

Tactic:
intro i

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ ν : Measure α
s t : Set α
a : α
μ : Measure α
inst✝ : SigmaFinite μ
c : ℝ≥0
i : ℕ
⊢ (c • μ) (spanningSets μ i) < ∞
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
⊢ ∀ (i : ι), (μ i) (s i) = 0

Tactic:
intro i

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
i : ι
⊢ (μ i) (s i) = 0
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
⊢ ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x

Tactic:
intro i

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
i : ι
⊢ ∀ x ∉ s i, f x = mk f ⋯ x
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_2.refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
n : ℕ
hn : n ∈ {k | ν (t' k) < μ (t' k)}
⊢ ν (t' ↑⟨n, hn⟩) = min (μ (t' ↑⟨n, hn⟩)) (ν (t' ↑⟨n, hn⟩))

Tactic:
mem_ofPred_eq

Goal state after:
case refine_2.refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
n : ℕ
hn✝ : n ∈ {k | ν (t' k) < μ (t' k)}
hn : ν (t' n) < μ (t' n)
⊢ ν (t' ↑⟨n, hn✝⟩) = min (μ (t' ↑⟨n, hn✝⟩)) (ν (t' ↑⟨n, hn✝⟩))
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
k : ℕ
hk₁ : k ∈ {k | μ (t' k) ≤ ν (t' k)}
hk₂ : k ∈ {k | ν (t' k) < μ (t' k)}
⊢ k ∈ ∅

Tactic:
mem_ofPred_eq

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t' : ℕ → Set α
ht' : s ⊆ iUnion t'
t : Set α := ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
ht : t = ⋃ n ∈ {k | μ (t' k) ≤ ν (t' k)}, t' n
hle₁ : μ (t ∩ s) ≤ ∑' (n : ↑{k | μ (t' k) ≤ ν (t' k)}), μ (t' ↑n)
hcap : tᶜ ∩ s ⊆ ⋃ n ∈ {k | ν (t' k) < μ (t' k)}, t' n
hle₂ : ν (tᶜ ∩ s) ≤ ∑' (n : ↑{k | ν (t' k) < μ (t' k)}), ν (t' ↑n)
heq : {k | μ (t' k) ≤ ν (t' k)} ∪ {k | ν (t' k) < μ (t' k)} = univ
k : ℕ
hk₁ : μ (t' k) ≤ ν (t' k)
hk₂ : ν (t' k) < μ (t' k)
⊢ k ∈ ∅
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
⊢ Measurable fun p ↦ p.1 / p.2

Tactic:
by
    simp only [div_eq_mul_inv]
    exact measurable_fst.mul measurable_snd.inv

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

Goal state before:
pp: case h₁
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
hα : SecondCountableTopology α
F : Set α → Set β := fun a ↦ {y | ∃ b, IsOpen[inst✝⁹] b ∧ y ∈ b ∧ a ×ˢ b ⊆ s}
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
y1 : α
y2 : β
hy : (y1, y2) ∈ s
u : Set α
v : Set β
u_open : IsOpen[inst✝¹²] u
v_open : IsOpen[inst✝⁹] v
yu : y1 ∈ u
yv : y2 ∈ v
huv : u ×ˢ v ⊆ s
a : Set α
ha : a ∈ countableBasis α
ya : y1 ∈ a
au : a ⊆ u
⊢ (y1, y2) ∈ ⋃ a ∈ countableBasis α, a ×ˢ F a

Tactic:
simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop]

Goal state after:
case h₁
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
hα : SecondCountableTopology α
F : Set α → Set β := fun a ↦ {y | ∃ b, IsOpen[inst✝⁹] b ∧ y ∈ b ∧ a ×ˢ b ⊆ s}
A : ∀ (a : Set α), IsOpen[inst✝⁹] (F a)
y1 : α
y2 : β
hy : (y1, y2) ∈ s
u : Set α
v : Set β
u_open : IsOpen[inst✝¹²] u
v_open : IsOpen[inst✝⁹] v
yu : y1 ∈ u
yv : y2 ∈ v
huv : u ×ˢ v ⊆ s
a : Set α
ha : a ∈ countableBasis α
ya : y1 ∈ a
au : a ⊆ u
⊢ ∃ i, y1 ∈ i ∧ i ∈ countableBasis α ∧ y2 ∈ F i
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case h₁
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
F : Set β → Set α := fun a ↦ {y | ∃ b, IsOpen[inst✝¹²] b ∧ y ∈ b ∧ b ×ˢ a ⊆ s}
A : ∀ (a : Set β), IsOpen[inst✝¹²] (F a)
y1 : α
y2 : β
hy : (y1, y2) ∈ s
u : Set α
v : Set β
u_open : IsOpen[inst✝¹²] u
v_open : IsOpen[inst✝⁹] v
yu : y1 ∈ u
yv : y2 ∈ v
huv : u ×ˢ v ⊆ s
a : Set β
ha : a ∈ countableBasis β
ya : y2 ∈ a
au : a ⊆ v
⊢ (y1, y2) ∈ ⋃ a ∈ countableBasis β, F a ×ˢ a

Tactic:
simp only [mem_iUnion, mem_prod, exists_and_left, exists_prop]

Goal state after:
case h₁
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
F : Set β → Set α := fun a ↦ {y | ∃ b, IsOpen[inst✝¹²] b ∧ y ∈ b ∧ b ×ˢ a ⊆ s}
A : ∀ (a : Set β), IsOpen[inst✝¹²] (F a)
y1 : α
y2 : β
hy : (y1, y2) ∈ s
u : Set α
v : Set β
u_open : IsOpen[inst✝¹²] u
v_open : IsOpen[inst✝⁹] v
yu : y1 ∈ u
yv : y2 ∈ v
huv : u ×ˢ v ⊆ s
a : Set β
ha : a ∈ countableBasis β
ya : y2 ∈ a
au : a ⊆ v
⊢ ∃ i, y1 ∈ F i ∧ i ∈ countableBasis β ∧ y2 ∈ i
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case refine_1
α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
x : α
xs : x ∈ s
y : α
yt : y ∈ t
z : α
hz : z ∈ s
⊢ ∃ y ∈ t, edist z y ≤ ediam (s ∪ t)

Tactic:
exact ⟨y, yt, Metric.edist_le_ediam_of_mem (subset_union_left hz) (subset_union_right yt)⟩

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

Goal state before:
pp: 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✝¹¹ : TopologicalSpace α✝
mα : MeasurableSpace α✝
inst✝¹⁰ : BorelSpace α✝
mβ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
inst✝³ : TopologicalSpace α
inst✝² : DiscreteTopology α
inst✝¹ : MeasurableSpace α
inst✝ : DiscreteMeasurableSpace α
⊢ BorelSpace α

Tactic:
by
  constructor; ext; simp [MeasurableSpace.measurableSet_generateFrom, MeasurableSet.of_discrete]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
hx : x ∈ {x | infEDist x E < ENNReal.ofReal δ}
⊢ x ∈ cthickening δ E

Tactic:
mem_ofPred_eq

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
hx : infEDist x E < ENNReal.ofReal δ
⊢ x ∈ cthickening δ E
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x : α
hs : s.Nonempty
⊢ IsGLB ((fun x_1 ↦ dist x x_1) '' s) (infDist x s)

Tactic:
by
  simpa [infDist_eq_iInf, sInf_image']
    using isGLB_csInf (hs.image _) ⟨0, by simp [lowerBounds]⟩

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

Goal state before:
pp: case succ
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
a : α
n : ℕ
IH : ∑ k ∈ Finset.range (n + 1), ↑((eapproxDiff f k) a) = (eapprox f n) a
⊢ (eapprox f n) a + ↑((eapprox f (n + 1)) a - (eapprox f n) a).toNNReal = (eapprox f (n + 1)) a

Tactic:
ENNReal.coe_toNNReal,

Goal state after:
case succ
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
a : α
n : ℕ
IH : ∑ k ∈ Finset.range (n + 1), ↑((eapproxDiff f k) a) = (eapprox f n) a
⊢ (eapprox f n) a + ((eapprox f (n + 1)) a - (eapprox f n) a) = (eapprox f (n + 1)) a

case succ
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
a : α
n : ℕ
IH : ∑ k ∈ Finset.range (n + 1), ↑((eapproxDiff f k) a) = (eapprox f n) a
⊢ (eapprox f (n + 1)) a - (eapprox f n) a ≠ ∞
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
⊢ ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}

Tactic:
intro i

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
i : ι
⊢ ∀ (c : α), MeasurableSet {x | f i x ≤ c}
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: ι : Type u_1
inst✝² : Preorder ι
inst✝¹ : Nonempty ι
inst✝ : IsDirected ι fun x1 x2 ↦ x1 ≤ x2
f : ι → ℝ
x : ℝ
h_tto : Tendsto f atTop (𝓝 x)
h_ant : Antitone f
h_bdd : BddBelow (range f)
⊢ IsGLB (range f) x

Tactic:
by
  rw [tendsto_nhds_unique h_tto (tendsto_atTop_ciInf h_ant h_bdd)]
  exact isGLB_ciInf h_bdd

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

Goal state before:
pp: 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
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, AEMeasurable (f i) μ
g : ι → δ → α := fun i ↦ if hi : i ∈ s then mk (f i) ⋯ else fun _b ↦ sSup ∅
⊢ ∀ i ∈ s, Measurable (g i)

Tactic:
intro i

Goal state after:
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
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, AEMeasurable (f i) μ
g : ι → δ → α := fun i ↦ if hi : i ∈ s then mk (f i) ⋯ else fun _b ↦ sSup ∅
i : ι
⊢ i ∈ s → Measurable (g i)
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
⊢ f.FinMeasSupp μ → ∀ (y : β), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞

Tactic:
refine fun h y hy => lt_of_le_of_lt (measure_mono ?_) h

Goal state after:
case mp
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
h : f.FinMeasSupp μ
y : β
hy : y ≠ 0
⊢ ⇑f ⁻¹' {y} ⊆ {x | (fun x ↦ f x = 0 x) x}ᶜ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case hn
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
⊢ Pairwise (Function.onFun Disjoint fun n ↦ s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Tactic:
intro i

Goal state after:
case hn
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
i : ℤ
⊢ ∀ ⦃j : ℤ⦄, i ≠ j → Function.onFun Disjoint (fun n ↦ s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) i j
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : IsEmpty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c

Tactic:
simp [eq_iff_true_of_subsingleton]

Goal state after: