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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' s)

Tactic:
← sum_sfiniteSeq ν

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x ↦ (sum (sfiniteSeq ν)) (Prod.mk x ⁻¹' s)
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
s : Set M
hs : MeasurableSet s
h : ρ s = 0
⊢ Measurable (s.indicator 1)

Tactic:
by measurability

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
s : Set α
t : Set β
S : Set α := toMeasurable μ s
T : Set β := toMeasurable ν t
⊢ MeasurableSet (S ×ˢ T)

Tactic:
by measurability

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

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
⊢ MeasurableSet (range f)

Tactic:
have : ∀ p : A, ∃ q : Set β,
      f '' (p.1.1 : Set γ) ⊆ q ∧ Disjoint (f '' (p.1.2 : Set γ)) q ∧ MeasurableSet q := by
    intro p
    apply
      AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)
        ((hb.isOpen p.1.2.2).analyticSet_image f_cont)
    exact Disjoint.image p.2 f_inj.injOn (subset_univ _) (subset_univ _)

Goal state after:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
this : ∀ (p : A), ∃ q, f '' ↑(↑p).1 ⊆ q ∧ Disjoint (f '' ↑(↑p).2) q ∧ MeasurableSet q
⊢ MeasurableSet (range f)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
μ✝ μ' : Measure α✝
ν✝ ν' : Measure β✝
τ : Measure γ
inst✝² : SFinite ν✝
α : Type u_4
β : Type u_5
x✝¹ : MeasurableSpace α
μ : Measure α
inst✝¹ : SFinite μ
x✝ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
| (sum (sfiniteSeq μ)).prod ν

Tactic:
← sum_sfiniteSeq ν

Goal state after:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
μ✝ μ' : Measure α✝
ν✝ ν' : Measure β✝
τ : Measure γ
inst✝² : SFinite ν✝
α : Type u_4
β : Type u_5
x✝¹ : MeasurableSpace α
μ : Measure α
inst✝¹ : SFinite μ
x✝ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
| (sum (sfiniteSeq μ)).prod (sum (sfiniteSeq ν))
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case refine_2
α : Type u_4
β : Type u_5
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure (α × β)
inst✝ : IsFiniteMeasure μ
h : ∀ {s : Set α} {t : Set β}, MeasurableSet s → MeasurableSet t → μ (s ×ˢ t) = ν (s ×ˢ t)
s : Set (α × β)
hs : MeasurableSet s
h_univ : μ univ = ν univ
this : IsFiniteMeasure ν
⊢ ∀ (t : Set (α × β)), MeasurableSet t → μ t = ν t → μ tᶜ = ν tᶜ

Tactic:
intro t ht h

Goal state after:
case refine_2
α : Type u_4
β : Type u_5
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure (α × β)
inst✝ : IsFiniteMeasure μ
h✝ : ∀ {s : Set α} {t : Set β}, MeasurableSet s → MeasurableSet t → μ (s ×ˢ t) = ν (s ×ˢ t)
s : Set (α × β)
hs : MeasurableSet s
h_univ : μ univ = ν univ
this : IsFiniteMeasure ν
t : Set (α × β)
ht : MeasurableSet t
h : μ t = ν t
⊢ μ tᶜ = ν tᶜ
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case refine_1
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure (α × β × γ)
inst✝ : IsFiniteMeasure μ
h :
  ∀ {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s → MeasurableSet t → MeasurableSet u → μ (s ×ˢ t ×ˢ u) = ν (s ×ˢ t ×ˢ u)
s : Set (α × β × γ)
hs : MeasurableSet s
h_univ : μ univ = ν univ
this : IsFiniteMeasure ν
C₂ : Set (Set (β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {t | MeasurableSet t} {u | MeasurableSet u}
C : Set (Set (α × β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} C₂
⊢ Prod.instMeasurableSpace = generateFrom C

Tactic:
refine (generateFrom_eq_prod (C := { s : Set α | MeasurableSet s }) (D := C₂) (by simp)
      generateFrom_prod isCountablySpanning_measurableSet ?_).symm
    exact isCountablySpanning_measurableSet.prod isCountablySpanning_measurableSet

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 ↔ LeftCosetEquivalence (↑K) g₁ g₂

Tactic:
rw [LeftCosetEquivalence, leftCoset_eq_iff]

Goal state after:
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 ↔ g₁⁻¹ * g₂ ∈ K
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
⊢ LeftCosetEquivalence (↑K) g ↑(hSK.equiv g).1

Tactic:
rw [equiv_fst_eq_mul_inv]; simp [LeftCosetEquivalence, leftCoset_eq_iff]

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

Goal state before:
pp: case refine_4
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure (α × β × γ)
inst✝ : IsFiniteMeasure μ
h :
  ∀ {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s → MeasurableSet t → MeasurableSet u → μ (s ×ˢ t ×ˢ u) = ν (s ×ˢ t ×ˢ u)
s : Set (α × β × γ)
hs : MeasurableSet s
h_univ : μ univ = ν univ
this : IsFiniteMeasure ν
C₂ : Set (Set (β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {t | MeasurableSet t} {u | MeasurableSet u}
C : Set (Set (α × β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} C₂
⊢ ∀ (t : Set (α × β × γ)), MeasurableSet t → μ t = ν t → μ tᶜ = ν tᶜ

Tactic:
intro t ht h

Goal state after:
case refine_4
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure (α × β × γ)
inst✝ : IsFiniteMeasure μ
h✝ :
  ∀ {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s → MeasurableSet t → MeasurableSet u → μ (s ×ˢ t ×ˢ u) = ν (s ×ˢ t ×ˢ u)
s : Set (α × β × γ)
hs : MeasurableSet s
h_univ : μ univ = ν univ
this : IsFiniteMeasure ν
C₂ : Set (Set (β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {t | MeasurableSet t} {u | MeasurableSet u}
C : Set (Set (α × β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} C₂
t : Set (α × β × γ)
ht : MeasurableSet t
h : μ t = ν t
⊢ μ tᶜ = ν tᶜ
Module: Mathlib.MeasureTheory.Group.Action

Goal state before:
pp: G : Type u
α : Type w
m : MeasurableSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
c : G
s t : Set α
⊢ μ (s ∪ c⁻¹ • t) = μ (c • s ∪ t)

Tactic:
rw [← measure_smul _ c, smul_set_union, smul_smul, mul_inv_cancel, one_smul]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
⊢ MeasurableSet F → ∀ r < μ F, ∃ K ⊆ F, p K ∧ r < μ K

Tactic:
hF

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
⊢ ∀ r < μ F, ∃ K ⊆ F, p K ∧ r < μ K
Module: Mathlib.MeasureTheory.Group.Action

Goal state before:
pp: G : Type u
α : Type w
m : MeasurableSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
c : G
⊢ c • ae μ = ae μ

Tactic:
ext s

Goal state after:
G : Type u
α : Type w
m : MeasurableSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
c : G
s : Set α
⊢ s ∈ c • ae μ ↔ s ∈ ae μ
Module: Mathlib.MeasureTheory.Group.Action

Goal state before:
pp: M : Type uM
α : Type uα
β : Type uβ
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : SMul M α
inst✝² : SMul M β
inst✝¹ : MeasurableConstSMul M β
μ : Measure α
inst✝ : SMulInvariantMeasure M α μ
f : α → β
hsmul : ∀ (m : M) (a : α), f (m • a) = m • f a
hf : Measurable f
m : M
S : Set β
hS : MeasurableSet S
⊢ μ ((fun x ↦ m • x) ⁻¹' f ⁻¹' S) = μ (f ⁻¹' S)

Tactic:
rw [SMulInvariantMeasure.measure_preimage_smul m (hS.preimage hf)]

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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
p : Set α → Prop
inst✝¹ : TopologicalSpace α
inst✝ : μ.OuterRegular
H : μ.InnerRegularWRT p IsOpen[inst✝¹]
hd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \ U)
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
h0 : p ∅
ε : ℝ≥0∞
hε : ε ≠ 0
hεs : ε + ε ≤ μ s
hr : μ s - (ε + ε) < μ s
U : Set α
hsU : U ⊇ s
hUo : IsOpen[inst✝¹] U
hUt : μ U < ∞
hμU : μ (U \ s) < ε
U' : Set α
hU'o : IsOpen[inst✝¹] U'
hμU' : μ U' < ε
hsU' : U \ U' ⊆ s
K : Set α
hKU : K ⊆ U
hKc : p K
hKr : μ U < μ K + ε
⊢ μ K + ε ≤ μ (K \ U') + μ U' + ε

Tactic:
grw [tsub_le_iff_right.1 le_measure_sdiff]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y ↦ f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ (μ.prod ν) (f ⁻¹' s) = 0

Tactic:
Measure.prod_apply (hf hs)

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y ↦ f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' f ⁻¹' s) ∂μ = 0
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case h₁
α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := ⋯
hu : u.Countable
t' : Set α := ⋯
ht' : MeasurableSet t'
this : g '' t = g '' (t \ t') ∪ g '' t'
⊢ InjOn g (t \ t')

Tactic:
intro x hx y hy hxy

Goal state after:
case h₁
α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := {c | ∃ x y, x ∈ t ∧ y ∈ t ∧ x < y ∧ g x = c ∧ g y = c}
hu : u.Countable
t' : Set α := t ∩ g ⁻¹' u
ht' : MeasurableSet t'
this : g '' t = g '' (t \ t') ∪ g '' t'
x : α
hx : x ∈ t \ t'
y : α
hy : y ∈ t \ t'
hxy : g x = g y
⊢ x = y
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝² : MeasurableSpace G
inst✝¹ : DivInvMonoid G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
g : G
⊢ Measure.map (fun x ↦ x / g) μ = μ

Tactic:
simp_rw [div_eq_mul_inv, map_mul_right_eq_self μ g⁻¹]

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

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSpace H
inst✝³ : DivisionMonoid G
inst✝² : MeasurableMul G
inst✝¹ : MeasurableInv G
μ : Measure G
inst✝ : μ.IsMulLeftInvariant
⊢ ∀ (g : G), map (fun x ↦ x * g) μ.inv = μ.inv

Tactic:
intro g

Goal state after:
G : Type u_1
H : Type u_2
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSpace H
inst✝³ : DivisionMonoid G
inst✝² : MeasurableMul G
inst✝¹ : MeasurableInv G
μ : Measure G
inst✝ : μ.IsMulLeftInvariant
g : G
⊢ map (fun x ↦ x * g) μ.inv = μ.inv
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSpace H
inst✝³ : DivisionMonoid G
inst✝² : MeasurableMul G
inst✝¹ : MeasurableInv G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
⊢ ∀ (g : G), map (fun x ↦ g * x) μ.inv = μ.inv

Tactic:
intro g

Goal state after:
G : Type u_1
H : Type u_2
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSpace H
inst✝³ : DivisionMonoid G
inst✝² : MeasurableMul G
inst✝¹ : MeasurableInv G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
g : G
⊢ map (fun x ↦ g * x) μ.inv = μ.inv
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
ρ : Measure (α × β)
s : Set α
hs : MeasurableSet s
⊢ ρ.fst s = ρ (Prod.fst ⁻¹' s)

Tactic:
fst,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
ρ : Measure (α × β)
s : Set α
hs : MeasurableSet s
⊢ (map Prod.fst ρ) s = ρ (Prod.fst ⁻¹' s)
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ ∑' (n : ℕ), (μ (s n) + δ n) = μ (⋃ n, s n) + ∑' (n : ℕ), δ n

Tactic:
rw [measure_iUnion hsd hsm, ENNReal.tsum_add]

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
h : ρ = 0
⊢ IsZeroOrProbabilityMeasure ρ.fst

pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
h : IsProbabilityMeasure ρ
⊢ IsZeroOrProbabilityMeasure ρ.fst

Tactic:
· simp only [h, fst_zero]
    infer_instance

Goal state after:
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
h : IsProbabilityMeasure ρ
⊢ IsZeroOrProbabilityMeasure ρ.fst
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
⊢ IsClosed[inst✝²] F → ∀ r < μ F, ∃ K ⊆ F, IsCompact K ∧ r < μ K

Tactic:
hF

Goal state after:
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
hF : IsClosed[inst✝²] F
⊢ ∀ r < μ F, ∃ K ⊆ F, IsCompact K ∧ r < μ K
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
ρ : Measure (α × β)
⊢ (map Prod.swap ρ).snd = ρ.fst

Tactic:
rw [Measure.snd, Measure.map_map measurable_snd measurable_swap]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
ρ : Measure (α × β)
⊢ map (Prod.snd ∘ Prod.swap) ρ = ρ.fst
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
⊢ (μ.withDensity f).restrict s = (μ.restrict s).withDensity f

Tactic:
ext1 t ht
  rw [restrict_apply ht, withDensity_apply _ ht, withDensity_apply _ (ht.inter hs),
    restrict_restrict ht]

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

Goal state before:
pp: case refine_2
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
s : Set G
s_meas : MeasurableSet s
μs : μ s ≠ ∞
r : ℝ≥0∞
hr : r < μ s
K : Set G
Ks : K ⊆ s
K_comp : IsCompact K
hK : r < μ K
⊢ r < μ (closure[inst✝³] K)

Tactic:
K_comp.measure_closure

Goal state after:
case refine_2
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
s : Set G
s_meas : MeasurableSet s
μs : μ s ≠ ∞
r : ℝ≥0∞
hr : r < μ s
K : Set G
Ks : K ⊆ s
K_comp : IsCompact K
hK : r < μ K
⊢ r < μ K
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: case inl
G : Type u_1
H : Type u_2
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
μ✝ : Measure G
inst✝⁶ : μ✝.IsHaarMeasure
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : T1Space G
inst✝² : WeaklyLocallyCompactSpace G
inst✝¹ : (𝓝[≠] 1).NeBot
μ : Measure G
inst✝ : μ.IsHaarMeasure
h : μ 1 = 0
⊢ NullSingletonClass μ

Tactic:
| inl h =>

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

Goal state before:
pp: case inr
G : Type u_1
H : Type u_2
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
μ✝ : Measure G
inst✝⁶ : μ✝.IsHaarMeasure
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : T1Space G
inst✝² : WeaklyLocallyCompactSpace G
inst✝¹ : (𝓝[≠] 1).NeBot
μ : Measure G
inst✝ : μ.IsHaarMeasure
h : μ 1 ≠ 0
K : Set G
K_compact : IsCompact K
K_nhds : K ∈ 𝓝 1
K_inf : K.Infinite
⊢ NullSingletonClass μ

Tactic:
exact absurd (K_inf.meas_eq_top ⟨_, h, fun x _ ↦ (haar_singleton _ _).ge⟩)
      K_compact.measure_lt_top.ne

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

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
inst✝ : μ.WeaklyRegular
A : Set α
h'A : μ A ≠ ∞
this✝ : Fact (μ A < ∞)
V : Set α
V_open : IsOpen[inst✝²] V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
this : (μ.restrict A).InnerRegularWRT IsClosed[inst✝²] fun s ↦ MeasurableSet s
⊢ ∃ K ⊆ V, IsClosed[inst✝²] K ∧ r < (μ.restrict A) K

Tactic:
exact this V_open.measurableSet r hr

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

Goal state before:
pp: case add
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
⊢ ∀ ⦃f_1 g : α → ℝ≥0∞⦄,
    Disjoint (Function.support f_1) (Function.support g) →
      Measurable f_1 →
        Measurable g →
          ∫⁻ (a : α), f_1 a ∂μ.withDensity f = ∫⁻ (a : α), (f * f_1) a ∂μ →
            ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ →
              ∫⁻ (a : α), (f_1 + g) a ∂μ.withDensity f = ∫⁻ (a : α), (f * (f_1 + g)) a ∂μ

Tactic:
intro g

Goal state after:
case add
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g : α → ℝ≥0∞
⊢ ∀ ⦃g_1 : α → ℝ≥0∞⦄,
    Disjoint (Function.support g) (Function.support g_1) →
      Measurable g →
        Measurable g_1 →
          ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ →
            ∫⁻ (a : α), g_1 a ∂μ.withDensity f = ∫⁻ (a : α), (f * g_1) a ∂μ →
              ∫⁻ (a : α), (g + g_1) a ∂μ.withDensity f = ∫⁻ (a : α), (f * (g + g_1)) a ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case iSup
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
⊢ ∀ ⦃f_1 : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f_1 n)) →
      Monotone f_1 →
        (∀ (n : ℕ), ∫⁻ (a : α), f_1 n a ∂μ.withDensity f = ∫⁻ (a : α), (f * f_1 n) a ∂μ) →
          ∫⁻ (a : α), (fun x ↦ ⨆ n, f_1 n x) a ∂μ.withDensity f = ∫⁻ (a : α), (f * fun x ↦ ⨆ n, f_1 n x) a ∂μ

Tactic:
intro g

Goal state after:
case iSup
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g : ℕ → α → ℝ≥0∞
⊢ (∀ (n : ℕ), Measurable (g n)) →
    Monotone g →
      (∀ (n : ℕ), ∫⁻ (a : α), g n a ∂μ.withDensity f = ∫⁻ (a : α), (f * g n) a ∂μ) →
        ∫⁻ (a : α), (fun x ↦ ⨆ n, g n x) a ∂μ.withDensity f = ∫⁻ (a : α), (f * fun x ↦ ⨆ n, g n x) a ∂μ
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ x / √x / x = √x / x

Tactic:
div_sqrt

Goal state after:
x : ℝ
⊢ √x / x = √x / x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
hfm : Measurable f
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : Set α := {x | f x = ∞}
hs : MeasurableSet s
⊢ ∀ (i : ℕ), Measurable (s.indicator 1)

Tactic:
by measurability

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

Goal state before:
pp: case e_a.e_f
α : Type u_1
m0 : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
hfm : Measurable f
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : Set α := {x | f x = ∞}
hs : MeasurableSet s
x : α
⊢ s.indicator f x = (∑' (n : ℕ), s.indicator 1) x

Tactic:
rw [ENNReal.tsum_apply]

Goal state after:
case e_a.e_f
α : Type u_1
m0 : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
hfm : Measurable f
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : Set α := {x | f x = ∞}
hs : MeasurableSet s
x : α
⊢ s.indicator f x = ∑' (i : ℕ), s.indicator 1 x
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: n : ℕ
⊢ I ^ n = I ^ (n % 4)

Tactic:
conv_lhs => rw [← Nat.div_add_mod n 4]

Goal state after:
n : ℕ
⊢ I ^ (4 * (n / 4) + n % 4) = I ^ (n % 4)
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_3
V₂ : Type u_4
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
⊢ (f.comp (incl ⊤)).range = f.range

Tactic:
simp [comp_range, incl_range, ← AddMonoidHom.range_eq_map]; rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.ContinuousLinearMap

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Ring 𝕜₂
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : 𝕜 →ₗ[𝕜] E
x : 𝕜
⊢ ‖f (x * 1)‖ ≤ ‖f 1‖ * ‖x‖

Tactic:
rw [← smul_eq_mul, f.map_smul, mul_comm]

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Ring 𝕜₂
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : 𝕜 →ₗ[𝕜] E
x : 𝕜
⊢ ‖x • f 1‖ ≤ ‖x‖ * ‖f 1‖