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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
⊢ ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0

Tactic:
refine fun hu ↦ measure_mono_null ?_ (hf.preimage_null hu)

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
hu : ν (u ∩ t) = 0
⊢ f ⁻¹' u ∩ s ⊆ f ⁻¹' (u ∩ t)
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ∞
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
⊢ ∃ m, μ (⋂ n, ⋂ (_ : n ≤ m), f n) < ε

Tactic:
suffices Filter.Tendsto F Filter.atTop (𝓝 0) by
    let _ := hfin.nonempty
    rw [ENNReal.tendsto_atTop_zero_iff_lt_of_antitone hFAnti] at this
    exact this ε hε

Goal state after:
α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ∞
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
⊢ Tendsto F atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : CompactSpace α
⊢ IsFiniteMeasure μ ↔ IsFiniteMeasureOnCompacts μ

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : CompactSpace α
⊢ IsFiniteMeasure μ → IsFiniteMeasureOnCompacts μ

case mpr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : CompactSpace α
⊢ IsFiniteMeasureOnCompacts μ → IsFiniteMeasure μ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
R : Type u_8
inst✝⁴ : Semiring R
inst✝³ : IsDomain R
inst✝² : Module R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : Module.IsTorsionFree R ℝ≥0∞
c : R
hc : c ≠ 0
μ : Measure α
⊢ ae (c • μ) = ae μ

Tactic:
ext

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
R : Type u_8
inst✝⁴ : Semiring R
inst✝³ : IsDomain R
inst✝² : Module R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : Module.IsTorsionFree R ℝ≥0∞
c : R
hc : c ≠ 0
μ : Measure α
s✝ : Set α
⊢ s✝ ∈ ae (c • μ) ↔ s✝ ∈ ae μ
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
c : ℝ≥0∞
hc : c ≠ 0
μ : Measure α
⊢ ae (c • μ) = ae μ

Tactic:
ext

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
c : ℝ≥0∞
hc : c ≠ 0
μ : Measure α
s✝ : Set α
⊢ s✝ ∈ ae (c • μ) ↔ s✝ ∈ ae μ
Module: Mathlib.Order.Filter.CountableSeparatingOn

Goal state before:
pp: α : Type u_1
p : Set α → Prop
t : Set α
⊢ HasCountableSeparatingOn (↑t) (fun u ↦ ∃ v, p v ∧ Subtype.val ⁻¹' v = u) univ ↔ HasCountableSeparatingOn α p t

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
p : Set α → Prop
t : Set α
⊢ HasCountableSeparatingOn (↑t) (fun u ↦ ∃ v, p v ∧ Subtype.val ⁻¹' v = u) univ → HasCountableSeparatingOn α p t

case mpr
α : Type u_1
p : Set α → Prop
t : Set α
⊢ HasCountableSeparatingOn α p t → HasCountableSeparatingOn (↑t) (fun u ↦ ∃ v, p v ∧ Subtype.val ⁻¹' v = u) univ
Module: Mathlib.Order.Filter.CountableSeparatingOn

Goal state before:
pp: case refine_2.inl
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
hsl : s ∈ l
⊢ x ∈ s ↔ y ∈ s

Tactic:
| inl hsl =>

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

Goal state before:
pp: case refine_1.refine_1
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
n : ℕ
x✝ : MeasurableSet (natGeneratingSequence α n)
x : α
hx : x ∈ natGeneratingSequence α n
⊢ ∃ i, i n ∧ x ∈ countablyGeneratedAtom α i

pp: case refine_1.refine_2
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
n : ℕ
x✝ : MeasurableSet (natGeneratingSequence α n)
x : α
⊢ (∃ i, i n ∧ x ∈ countablyGeneratedAtom α i) → x ∈ natGeneratingSequence α n

Tactic:
· exact ⟨(x ∈ natGeneratingSequence α ·), by simp [countablyGeneratedAtom]; grind⟩

Goal state after:
case refine_1.refine_2
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
n : ℕ
x✝ : MeasurableSet (natGeneratingSequence α n)
x : α
⊢ (∃ i, i n ∧ x ∈ countablyGeneratedAtom α i) → x ∈ natGeneratingSequence α n
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set β
t : Set α
ht : MeasurableSet t
⊢ (μ.restrict s) (f '' t) = μ (f '' (t ∩ f ⁻¹' s))

Tactic:
Measure.restrict_apply (hf.measurableSet_image.2 ht),

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure β
s : Set β
t : Set α
ht : MeasurableSet t
⊢ μ (f '' t ∩ s) = μ (f '' (t ∩ f ⁻¹' s))
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
hμ : ∀ (a : α), μ {a} < ∞
⊢ SigmaFinite μ

Tactic:
by
    cases isEmpty_or_nonempty α
    · rw [Subsingleton.elim μ 0]
      infer_instance
    · obtain ⟨f, hf⟩ := exists_surjective_nat α
      exact ⟨⟨⟨fun n ↦ {f n}, by simp, by simpa [hf.forall] using hμ, by simp [hf.range_eq]⟩⟩⟩

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

Goal state before:
pp: α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
x✝ : MeasurableSpace β
f : α → β
g : β → α
μ : Measure α
hf : Measurable f
hg : MeasurableEmbedding g
hμg : ∀ᵐ (a : α) ∂μ, a ∈ range g
hfg : ∀ (a : β), f (g a) = a
⊢ map f μ = comap g μ

Tactic:
by
  ext s hs
  rw [map_apply hf hs, hg.comap_apply, ← measure_sdiff_null hμg]
  congr
  simp
  grind

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

Goal state before:
pp: case refine_3
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
q : (ℕ → Prop) → Prop
⊢ (t = ⋃ p, if q p then countablyGeneratedAtom α p else ∅) →
    ∃ q, tᶜ = ⋃ p, if q p then countablyGeneratedAtom α p else ∅

Tactic:
htq

Goal state after:
case refine_3
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
q : (ℕ → Prop) → Prop
htq : t = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
⊢ ∃ q, tᶜ = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_3.refine_1
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
q : (ℕ → Prop) → Prop
htq : t = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
x : α
h : ∀ (i : ℕ → Prop), q i → x ∉ countablyGeneratedAtom α i
⊢ ∃ i, ¬q i ∧ x ∈ countablyGeneratedAtom α i

Tactic:
refine ⟨fun n ↦ x ∈ natGeneratingSequence α n, ?_,
        mem_countablyGeneratedAtom_natGeneratingSequence x⟩

Goal state after:
case refine_3.refine_1
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
q : (ℕ → Prop) → Prop
htq : t = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
x : α
h : ∀ (i : ℕ → Prop), q i → x ∉ countablyGeneratedAtom α i
⊢ ¬q fun n ↦ x ∈ natGeneratingSequence α n
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_4
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
q : ℕ → (ℕ → Prop) → Prop
hq : ∀ (n : ℕ), t n = ⋃ p, if q n p then countablyGeneratedAtom α p else ∅
⊢ ⋃ i, t i = ⋃ p, if (fun p ↦ ⨆ n, q n p) p then countablyGeneratedAtom α p else ∅

Tactic:
ext

Goal state after:
case refine_4
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
q : ℕ → (ℕ → Prop) → Prop
hq : ∀ (n : ℕ), t n = ⋃ p, if q n p then countablyGeneratedAtom α p else ∅
x✝ : α
⊢ x✝ ∈ ⋃ i, t i ↔ x✝ ∈ ⋃ p, if (fun p ↦ ⨆ n, q n p) p then countablyGeneratedAtom α p else ∅
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 : α → β
inst✝ : Zero β
g : α → β
hs : MeasurableSet s
h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x
x : α
hx : x ∈ s → f x = g x
⊢ s.indicator f x = s.indicator g x

pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
g : α → β
hs : MeasurableSet s
h : s.indicator f =ᵐ[μ] s.indicator g
x : α
hx : s.indicator f x = s.indicator g x
⊢ x ∈ s → f x = g x

Tactic:
· by_cases hxs : x ∈ s
    · simp [hxs, hx hxs]
    · simp [hxs]

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
g : α → β
hs : MeasurableSet s
h : s.indicator f =ᵐ[μ] s.indicator g
x : α
hx : s.indicator f x = s.indicator g x
⊢ x ∈ s → f x = g x
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case refine_1
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
n : ℕ
⊢ μ (f ⁻¹' (f '' spanningSets μ n ∪ (range f)ᶜ)) < ∞

Tactic:
Set.preimage_union

Goal state after:
case refine_1
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
n : ℕ
⊢ μ (f ⁻¹' f '' spanningSets μ n ∪ f ⁻¹' (range f)ᶜ) < ∞
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case refine_2
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)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
t : Set β
ht : MeasurableSet t
x : α
hx : ∃ i, x ∉ s i
⊢ (∃ i, mk f ⋯ x ∈ t ∧ x ∉ s i) ↔ f x ∈ t

Tactic:
constructor

Goal state after:
case refine_2.mp
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 := ⋯
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := ⋯
t : Set β
ht : MeasurableSet t
x : α
hx : ∃ i, x ∉ s i
⊢ (∃ i, mk f ⋯ x ∈ t ∧ x ∉ s i) → f x ∈ t

case refine_2.mpr
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 := ⋯
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := ⋯
t : Set β
ht : MeasurableSet t
x : α
hx : ∃ i, x ∉ s i
⊢ f x ∈ t → ∃ i, mk f ⋯ x ∈ t ∧ x ∉ s i
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν : Measure α
f : α → β
hμ : AEMeasurable f μ
hν : AEMeasurable f ν
⊢ Measure.map f (μ + ν) = Measure.map f μ + Measure.map f ν

Tactic:
ext

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ ν : Measure α
f : α → β
hμ : AEMeasurable f μ
hν : AEMeasurable f ν
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ (Measure.map f (μ + ν)) s✝ = (Measure.map f μ + Measure.map f ν) s✝
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
μ : Measure α
inst✝ : Zero β
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable (s.indicator f) μ ↔ AEMeasurable f (μ.restrict s)

Tactic:
constructor

Goal state after:
case mp
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
μ : Measure α
inst✝ : Zero β
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable (s.indicator f) μ → AEMeasurable f (μ.restrict s)

case mpr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
μ : Measure α
inst✝ : Zero β
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable f (μ.restrict s) → AEMeasurable (s.indicator f) μ
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
⊢ AEMeasurable (s.indicator fun x ↦ b) μ ↔ NullMeasurableSet s μ

Tactic:
constructor

Goal state after:
case mp
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
⊢ AEMeasurable (s.indicator fun x ↦ b) μ → NullMeasurableSet s μ

case mpr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
⊢ NullMeasurableSet s μ → AEMeasurable (s.indicator fun x ↦ b) μ
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: case h
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
⊢ s ⊆ ⋃ n, (fun n ↦ (closure[inst✝¹] (g' n))ᶜ) n ∧
    ∀ (n : ℕ),
      IsOpen[inst✝¹] ((fun n ↦ (closure[inst✝¹] (g' n))ᶜ) n) ∧
        Disjoint (closure[inst✝¹] ((fun n ↦ (closure[inst✝¹] (g' n))ᶜ) n)) t

Tactic:
constructor

Goal state after:
case h.left
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
⊢ s ⊆ ⋃ n, (closure[inst✝¹] (g' n))ᶜ

case h.right
X : Type u_1
inst✝¹ : TopologicalSpace X
s t : Set X
inst✝ : NormalSpace X
st_dis : Disjoint s t
t_cl : IsClosed[inst✝¹] t
T : Set (Set X)
T_open : ∀ t ∈ T, IsOpen[inst✝¹] t
T_count : T.Countable
T_int : t = ⋂₀ T
T_nonempty : T.Nonempty
g : ℕ → ↑T
g_surj : Surjective g
g' : ℕ → Set X
g'_open : ∀ (n : ℕ), IsOpen[inst✝¹] (g' n)
clt_sub_g' : ∀ (n : ℕ), t ⊆ g' n
clg'_sub_g : ∀ (n : ℕ), closure[inst✝¹] (g' n) ⊆ ↑(g n)
clg'_int : t = ⋂ i, closure[inst✝¹] (g' i)
⊢ ∀ (n : ℕ), IsOpen[inst✝¹] (closure[inst✝¹] (g' n))ᶜ ∧ Disjoint (closure[inst✝¹] (closure[inst✝¹] (g' n))ᶜ) t
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α✝ : Type u_1
G : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m : MeasurableSpace α
mβ : MeasurableSpace β
f g : α → G
μ : Measure α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
⊢ MeasurableEq E

Tactic:
constructor

Goal state after:
α✝ : Type u_1
G : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Div G
m : MeasurableSpace α
mβ : MeasurableSpace β
f g : α → G
μ : Measure α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : AddGroup E
inst✝¹ : MeasurableSingletonClass E
inst✝ : MeasurableSub₂ E
⊢ MeasurableSet (Set.diagonal E)
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α✝ : Type u_1
G : Type u_2
α : Type u_3
β✝ : Type u_4
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Div G
m : MeasurableSpace α
mβ : MeasurableSpace β✝
f g : α → G
μ : Measure α
β : Type u_5
inst✝⁶ : AddCommMonoid β
inst✝⁵ : PartialOrder β
inst✝⁴ : CanonicallyOrderedAdd β
inst✝³ : Sub β
inst✝² : OrderedSub β
x✝ : MeasurableSpace β
inst✝¹ : MeasurableSub₂ β
inst✝ : MeasurableSingletonClass β
⊢ MeasurableEq β

Tactic:
constructor

Goal state after:
α✝ : Type u_1
G : Type u_2
α : Type u_3
β✝ : Type u_4
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Div G
m : MeasurableSpace α
mβ : MeasurableSpace β✝
f g : α → G
μ : Measure α
β : Type u_5
inst✝⁶ : AddCommMonoid β
inst✝⁵ : PartialOrder β
inst✝⁴ : CanonicallyOrderedAdd β
inst✝³ : Sub β
inst✝² : OrderedSub β
x✝ : MeasurableSpace β
inst✝¹ : MeasurableSub₂ β
inst✝ : MeasurableSingletonClass β
⊢ MeasurableSet (Set.diagonal β)
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✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
X : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
⊢ OpensMeasurableSpace ((i : ι) → X i)

Tactic:
constructor

Goal state after:
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✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : MeasurableSpace α
inst✝¹⁴ : OpensMeasurableSpace α
inst✝¹³ : TopologicalSpace β
inst✝¹² : MeasurableSpace β
inst✝¹¹ : OpensMeasurableSpace β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : TopologicalSpace γ₂
inst✝⁶ : MeasurableSpace γ₂
inst✝⁵ : BorelSpace γ₂
inst✝⁴ : MeasurableSpace δ
ι : Type u_6
X : ι → Type u_7
inst✝³ : Countable ι
t' : (i : ι) → TopologicalSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
⊢ borel ((i : ι) → X i) ≤ MeasurableSpace.pi
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 = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc 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 : ∀ b < a, (Ioo b a).Nonempty
y : α
⊢ y < a ↔ ∃ i, ∃ (_ : i ∈ t), ∃ i_1, ∃ (_ : i_1 ∈ t) (_ : i < i_1) (_ : i_1 < a), i ≤ y ∧ y ≤ i_1

Tactic:
constructor

Goal state after:
case mp
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 = ∅ → x ∈ s
S : Set (Set α) := ⋯
this : MeasurableSpace α := ⋯
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
y : α
⊢ y < a → ∃ i, ∃ (_ : i ∈ t), ∃ i_1, ∃ (_ : i_1 ∈ t) (_ : i < i_1) (_ : i_1 < a), i ≤ y ∧ y ≤ i_1

case mpr
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 = ∅ → x ∈ s
S : Set (Set α) := ⋯
this : MeasurableSpace α := ⋯
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
y : α
⊢ (∃ i, ∃ (_ : i ∈ t), ∃ i_1, ∃ (_ : i_1 ∈ t) (_ : i < i_1) (_ : i_1 < a), i ≤ y ∧ y ≤ i_1) → y < a
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 = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc 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
b : α
ba : b < a
hb : Ioo b a = ∅
hbs : b ∈ s
x : α
h : x < a
⊢ ∃ i ≤ x, i ≤ b ∧ i ∈ t ∧ x ≤ b

Tactic:
have : x ≤ b := by
        contrapose! hb
        exact ⟨x, hb, h⟩

Goal state after:
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 = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc 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
b : α
ba : b < a
hb : Ioo b a = ∅
hbs : b ∈ s
x : α
h : x < a
this : x ≤ b
⊢ ∃ i ≤ x, i ≤ b ∧ i ∈ t ∧ x ≤ b
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case inl
α : Type u
inst✝ : PseudoEMetricSpace α
t : Set α
ht : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t
hs : IsCompact ∅
hst : Disjoint ∅ t
⊢ ∃ r, 0 < r ∧ ∀ x ∈ ∅, ∀ y ∈ t, ↑r < edist x y

Tactic:
use 1
    simp

Goal state after: