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

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ₁ μ₂ ν : Measure α
h : μ₁ + μ₂ ≪ ν
⊢ μ₁ ≪ ν ∧ μ₂ ≪ ν

Tactic:
have : ∀ s, ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0 := by intro s hs0; simpa using h hs0
    exact ⟨fun s hs0 ↦ (this s hs0).1, fun s hs0 ↦ (this s hs0).2⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
hs : NullMeasurableSet s (comap f μ)
⊢ NullMeasurableSet (f '' s) μ

Tactic:
refine ⟨toMeasurable μ (f '' toMeasurable (μ.comap f) s), measurableSet_toMeasurable _ _, ?_⟩

Goal state after:
α : Type u_1
β : Type u_2
s : Set α
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
hs : NullMeasurableSet s (comap f μ)
⊢ f '' s =ᵐ[μ] toMeasurable μ (f '' toMeasurable (comap f μ) s)
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
⊢ comap (fun x ↦ x) μ = μ

Tactic:
ext s hs

Goal state after:
β : Type u_2
mβ : MeasurableSpace β
μ : Measure β
s : Set β
hs : MeasurableSet s
⊢ (comap (fun x ↦ x) μ) s = μ s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
f : α → β
g : β → γ
hf' : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)
hg : Injective g
hg' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (g '' s)
μ : Measure γ
hf : Injective f
⊢ comap f (comap g μ) = comap (g ∘ f) μ

Tactic:
ext s hs

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
f : α → β
g : β → γ
hf' : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)
hg : Injective g
hg' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (g '' s)
μ : Measure γ
hf : Injective f
s : Set α
hs : MeasurableSet s
⊢ (comap f (comap g μ)) s = (comap (g ∘ f) μ) s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
⊢ comap f (c • μ) = c • comap f μ

Tactic:
ext s hs

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
c : ℝ≥0∞
hc : c ≠ 0
h : Injective f ∧ ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s : Set α
hs : MeasurableSet s
⊢ (comap f (c • μ)) s = (c • comap f μ) s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure β
⊢ Measure.comap f (μ + ν) = Measure.comap f μ + Measure.comap f ν

Tactic:
ext s hs

Goal state after:
α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure β
s : Set α
hs : MeasurableSet s
⊢ (Measure.comap f (μ + ν)) s = (Measure.comap f μ + Measure.comap f ν) s
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
⊢ Measure.comap (⇑e.symm) μ = Measure.map (⇑e) μ

Tactic:
ext s hs

Goal state after:
α : Type u_1
β : Type u_2
ma : MeasurableSpace α
mb : MeasurableSpace β
μ : Measure α
e : α ≃ᵐ β
s : Set β
hs : MeasurableSet s
⊢ (Measure.comap (⇑e.symm) μ) s = (Measure.map (⇑e) μ) s
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
⊢ ⋃ i, spanningSets μ i = univ

Tactic:
spanningSets,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
⊢ ⋃ i, accumulate μ.toFiniteSpanningSetsIn.set i = univ
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)
⊢ (map f μ) univ < ∞

Tactic:
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 ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ ≤ ν
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
x✝ : s.Nonempty
⊢ ν s ≤ μ s

Tactic:
by_contra! h_lt

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ ≤ ν
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
x✝ : s.Nonempty
h_lt : μ s < ν s
⊢ False
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : Disjoint s t
hs : MeasurableSet s
⊢ μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t

Tactic:
rw [union_comm, restrict_union h.symm hs, add_comm]

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

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_8
inst✝³ : LinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ∞
b : ι
hab : a ⋖ b
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))

Tactic:
simp [hab.nhdsGT]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
⊢ μ = ν

Tactic:
ext s hs

Goal state after:
α : Type u_1
m : MeasurableSpace α
inst✝ : Countable α
μ ν : Measure α
h : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)
s : Set α
hs : MeasurableSet s
⊢ μ s = ν s
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h : μ t = μ u
htu : t ⊆ u
ht_ne_top : μ u ≠ ∞
⊢ μ (u ∩ s) ≤ μ (t ∩ s)

Tactic:
have A : μ (u ∩ s) + μ (u \ s) ≤ μ (t ∩ s) + μ (u \ s) :=
    calc
      μ (u ∩ s) + μ (u \ s) = μ u := measure_inter_add_sdiff _ hs
      _ = μ t := h.symm
      _ = μ (t ∩ s) + μ (t \ s) := (measure_inter_add_sdiff _ hs).symm
      _ ≤ μ (t ∩ s) + μ (u \ s) := by gcongr

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h : μ t = μ u
htu : t ⊆ u
ht_ne_top : μ u ≠ ∞
A : μ (u ∩ s) + μ (u \ s) ≤ μ (t ∩ s) + μ (u \ s)
⊢ μ (u ∩ s) ≤ μ (t ∩ s)
Module: Mathlib.MeasureTheory.Measure.Trim

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 : Measurable f
⊢ Measure.map f (μ.trim ⋯) = Measure.map f μ

Tactic:
ext s hs

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (Measure.map f (μ.trim ⋯)) s = (Measure.map f μ) s
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
ms : MeasurableSpace α
h : ms ≤ OuterMeasure.caratheodory 0
⊢ OuterMeasure.toMeasure 0 h = 0

Tactic:
ext s hs

Goal state after:
α : Type u_1
ms : MeasurableSpace α
h : ms ≤ OuterMeasure.caratheodory 0
s : Set α
hs : MeasurableSet s
⊢ (OuterMeasure.toMeasure 0 h) s = 0 s
Module: Mathlib.MeasureTheory.Measure.Trim

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
m₁ m₂ : MeasurableSpace α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
t : Set α
ht : MeasurableSet t
⊢ (μ.trim hm₂) t = (μ.trim ⋯) t

Tactic:
trim_measurableSet_eq (hm₁₂.trans hm₂) ht,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
m₁ m₂ : MeasurableSpace α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m0
t : Set α
ht : MeasurableSet t
⊢ (μ.trim hm₂) t = μ t
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
⊢ MeasurableSet u → (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ u

Tactic:
_hu

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
⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅ μ ∈ toOuterMeasure '' m, μ (t \ s) ≤ μ u
Module: Mathlib.Order.Filter.CountableSeparatingOn

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
inst✝ : CountableInterFilter l
f g : α → β
p : Set β → Prop
s : Set β
h' : HasCountableSeparatingOn β p s
hf : ∀ᶠ (x : α) in l, f x ∈ s
hg : ∀ᶠ (x : α) in l, g x ∈ s
h : ∀ (U : Set β), p U → ∀ᶠ (x : α) in l, f x ∈ U ↔ g x ∈ U
⊢ f =ᶠ[l] g

Tactic:
rcases h'.1 with ⟨S, hSc, hSp, hS⟩

Goal state after:
α : Type u_1
β : Type u_2
l : Filter α
inst✝ : CountableInterFilter l
f g : α → β
p : Set β → Prop
s : Set β
h' : HasCountableSeparatingOn β p s
hf : ∀ᶠ (x : α) in l, f x ∈ s
hg : ∀ᶠ (x : α) in l, g x ∈ s
h : ∀ (U : Set β), p U → ∀ᶠ (x : α) in l, f x ∈ U ↔ g x ∈ U
S : Set (Set β)
hSc : S.Countable
hSp : ∀ s ∈ S, p s
hS : ∀ x ∈ s, ∀ y ∈ s, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
⊢ f =ᶠ[l] g
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
ht : t ⊆ s
h : NullMeasurableSet t (μ.restrict s)
t' : Set α
t'_subs : t' ⊆ t
ht' : MeasurableSet t'
t't : t' =ᵐ[μ.restrict s] t
x : α
hx : t' x = t x
⊢ x ∈ t' ↔ x ∈ t

Tactic:
by simpa using! hx

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:
ext s hs

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
s : Set α
hs : MeasurableSet s
⊢ (ν₁.restrict (spanningSets μ i)) s = (ν₂.restrict (spanningSets μ i)) s
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:
ext s hs

Goal state after:
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
s : Set β
hs : MeasurableSet s
⊢ (map f μ) s = (comap g μ) s
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
⊢ ∑ i ∈ F, (μ.restrict (s i)) t ≤ (M • μ.restrict (⋃ i, s i)) t

Tactic:
let P (C : Finset ι) := (⋂ i ∈ C, s i) ∩ (⋂ i ∈ (F \ C), (s i)ᶜ)

Goal state after:
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
⊢ ∑ i ∈ F, (μ.restrict (s i)) t ≤ (M • μ.restrict (⋃ i, s i)) t
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_2
α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
x✝ : ∃ S, ↑S ⊆ memPartition t n ∧ s = ⋃₀ ↑S
S : Finset (Set α)
hS_subset : ↑S ⊆ memPartition t n
hS_eq : s = ⋃₀ ↑S
⊢ MeasurableSet s

Tactic:
· rw [hS_eq, sUnion_eq_biUnion]
    refine MeasurableSet.biUnion ?_ (fun t ht ↦ ?_)
    · exact S.countable_toSet
    · exact measurableSet_generateFrom (hS_subset ht)

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

Goal state before:
pp: case inr
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ : Measure α
h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y
hμ : μ ≠ 0
⊢ AEMeasurable f μ

Tactic:
have := ae_neBot.2 hμ
    rcases h.exists with ⟨x, hx⟩
    exact ⟨const α (f x), measurable_const, EventuallyEq.symm hx⟩

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

Goal state before:
pp: 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
h : NullMeasurableSet s μ
⊢ AEMeasurable (s.indicator fun x ↦ b) μ

Tactic:
exact (aemeasurable_indicator_iff₀ h).mpr aemeasurable_const

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

Goal state before:
pp: α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
⊢ (sum fun x ↦ 0) = 0

Tactic:
ext s hs

Goal state after:
α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
s : Set α
hs : MeasurableSet s
⊢ (sum fun x ↦ 0) s = 0 s
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
ι : Type u_8
ι' : Type u_9
e : ι' ≃ ι
m : ι → Measure α
⊢ sum (m ∘ ⇑e) = sum m

Tactic:
ext s hs

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
ι : Type u_8
ι' : Type u_9
e : ι' ≃ ι
m : ι → Measure α
s : Set α
hs : MeasurableSet s
⊢ (sum (m ∘ ⇑e)) s = (sum m) s
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
ι : Type u_8
ι' : Type u_9
f : ι → ι'
hf : Injective f
m : ι → Measure α
⊢ sum (Function.extend f m 0) = sum m

Tactic:
ext s hs

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
ι : Type u_8
ι' : Type u_9
f : ι → ι'
hf : Injective f
m : ι → Measure α
s : Set α
hs : MeasurableSet s
⊢ (sum (Function.extend f m 0)) s = (sum m) s
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: case refine_2
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 s : Set α
hs : s ∈ μ.cofinite
⊢ (toMeasurable μ sᶜ)ᶜ ⊆ s

Tactic:
rw [compl_subset_comm]

Goal state after:
case refine_2
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 s : Set α
hs : s ∈ μ.cofinite
⊢ sᶜ ⊆ toMeasurable μ sᶜ
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
ι : Type u_7
m : ι → Measure α
f : α → β
hf : AEMeasurable f (sum m)
⊢ map f (sum m) = sum fun i ↦ map f (m i)

Tactic:
ext s hs

Goal state after:
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
ι : Type u_7
m : ι → Measure α
f : α → β
hf : AEMeasurable f (sum m)
s : Set β
hs : MeasurableSet s
⊢ (map f (sum m)) s = (sum fun i ↦ map f (m i)) s
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 : MeasurableMul G
inst✝ : MeasurableInv G
c : G
⊢ (fun x ↦ c / x) = fun x ↦ c * x⁻¹

Tactic:
ext1

Goal state after:
α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul G
inst✝ : MeasurableInv G
c x✝ : G
⊢ c / x✝ = c * x✝⁻¹
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 : MeasurableMul G
inst✝ : MeasurableInv G
c : G
⊢ (fun x ↦ x / c) = fun x ↦ id x * c⁻¹

Tactic:
ext1

Goal state after:
α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul G
inst✝ : MeasurableInv G
c x✝ : G
⊢ x✝ / c = id x✝ * c⁻¹
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: case ofNat
α✝ : Type u_1
M✝ : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
inst✝⁷ : MeasurableSpace X
inst✝⁶ : SMul M✝ X
m : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → M✝
g : α → X
inst✝⁵ : MeasurableSpace M✝
inst✝⁴ : MeasurableSMul M✝ X
M : Type u_6
inst✝³ : SubNegMonoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableAdd₂ M
inst✝ : MeasurableNeg M
n : ℕ
⊢ Measurable fun x ↦ n • x

Tactic:
exact measurable_const_smul _

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

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)

Tactic:
have H : ∀ a : α, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ ⟨_, rfl⟩

Goal state after:
case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
r : ℝ≥0∞
H1 : ∀ x ∈ s, ∃ y ∈ t, edist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, edist x y ≤ r
x : α
xt : x ∈ t
y : α
ys : y ∈ s
hy : edist x y ≤ r
⊢ infEDist x s ≤ r

Tactic:
exact le_trans (infEDist_le_edist_of_mem ys) hy

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
x : α
xs : x ∈ s
⊢ infEDist x t + hausdorffEDist t u ≤ hausdorffEDist s t + hausdorffEDist t u

Tactic:
infEDist_le_hausdorffEDist_of_mem xs

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
x : α
xs : x ∈ s
⊢ hausdorffEDist s t + hausdorffEDist t u ≤ hausdorffEDist s t + hausdorffEDist t u
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
x : α
xu : x ∈ u
⊢ infEDist x t + hausdorffEDist t s ≤ hausdorffEDist u t + hausdorffEDist t s

Tactic:
infEDist_le_hausdorffEDist_of_mem xu

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
x : α
xu : x ∈ u
⊢ hausdorffEDist u t + hausdorffEDist t s ≤ hausdorffEDist u t + hausdorffEDist t s
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr
X : Type u
inst✝ : PseudoMetricSpace X
δ : ℝ
E : Set X
h : Bornology.IsBounded E
x : X
hx : x ∈ E
⊢ Bornology.IsBounded (thickening δ E)

Tactic:
· refine (isBounded_iff_subset_closedBall x).2 ⟨δ + diam E, fun y hy ↦ ?_⟩
    calc
      dist y x ≤ infDist y E + diam E := dist_le_infDist_add_diam (x := y) h hx
      _ ≤ δ + diam E := by grw [(mem_thickening_iff_infDist_lt ⟨x, hx⟩).1 hy]

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 ∈ thickening δ E
⊢ x ∈ cthickening δ E

Tactic:
rw [thickening, mem_ofPred_eq] at hx

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.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : μ univ = ν univ
h : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ μ (⇑OrderDual.ofDual ⁻¹' Ioc b a) = ν (⇑OrderDual.ofDual ⁻¹' Ioc b a)

Tactic:
exact h hab

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
μ : Measure α
⊢ ∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (⇑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (⇑f ⁻¹' {x})

Tactic:
refine finsetSum_iSup fun i j ↦ (hd i j).imp fun k ⟨hik, hjk⟩ ↦ fun a ↦ ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
μ : Measure α
i j k : ι
x✝ : (fun x1 x2 ↦ x1 ⊆ x2) (s i) (s k) ∧ (fun x1 x2 ↦ x1 ⊆ x2) (s j) (s k)
hik : (fun x1 x2 ↦ x1 ⊆ x2) (s i) (s k)
hjk : (fun x1 x2 ↦ x1 ⊆ x2) (s j) (s k)
a : ℝ≥0∞
⊢ a * (μ.restrict (s i)) (⇑f ⁻¹' {a}) ≤ a * (μ.restrict (s k)) (⇑f ⁻¹' {a}) ∧
    a * (μ.restrict (s j)) (⇑f ⁻¹' {a}) ≤ a * (μ.restrict (s k)) (⇑f ⁻¹' {a})
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:
exact measurable_const_smul _

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
⊢ Measurable g

Tactic:
rcases u.exists_seq_tendsto with ⟨x, hx⟩

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
x : ℕ → ι
hx : Tendsto x atTop u
⊢ Measurable g
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
this : Nonempty ↑s
hx :
  f x ∈
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace]
      (Subtype.val '' closure[instTopologicalSpaceSubtype] (range (denseSeq ↑s)))
⊢ Tendsto (fun n ↦ (⇑(nearestPt (fun k ↦ Nat.casesOn k y₀ (Subtype.val ∘ denseSeq ↑s)) n) ∘ f) x) atTop (𝓝 (f x))

Tactic:
refine tendsto_nearestPt (closure_minimal ?_ isClosed_closure hx)

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
this : Nonempty ↑s
hx :
  f x ∈
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace]
      (Subtype.val '' closure[instTopologicalSpaceSubtype] (range (denseSeq ↑s)))
⊢ Subtype.val '' closure[instTopologicalSpaceSubtype] (range (denseSeq ↑s)) ⊆
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace]
      (range fun k ↦ Nat.casesOn k y₀ (Subtype.val ∘ denseSeq ↑s))