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

Goal state before:
pp: case refine_1
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ f ⁻¹' K ⊆ U

pp: case refine_2
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ r < (comap f μ) (f ⁻¹' K)

Tactic:
·

Goal state after:
case refine_1
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ f ⁻¹' K ⊆ U
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_2
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement S ↑H''
⊢ Nat.card ↑(⇑H.subtype '' S) * Nat.card ↥(map H.subtype H'') = Nat.card ↥H

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement S ↑H''
⊢ Nat.card ↑(⇑H.subtype '' S) * Nat.card ↥(map H.subtype H'') = Nat.card ↥H
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case refine_2
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ r < (comap f μ) (f ⁻¹' K)

Tactic:
·

Goal state after:
case refine_2
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ r < (comap f μ) (f ⁻¹' K)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case h₁.refine_2
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 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
y : γ
n : ℕ
s : Set γ
sb : s ∈ b
ys : y ∈ s
hs : s ⊆ ball y (u n / 2)
diam_s : diam s ≤ u n
t : ↑b
ht : Disjoint ↑⟨s, sb⟩ ↑t
⊢ f y ∉ q ⟨(t, ⟨s, sb⟩), ⋯⟩

Tactic:
·

Goal state after:
case h₁.refine_2
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 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
y : γ
n : ℕ
s : Set γ
sb : s ∈ b
ys : y ∈ s
hs : s ⊆ ball y (u n / 2)
diam_s : diam s ≤ u n
t : ↑b
ht : Disjoint ↑⟨s, sb⟩ ↑t
⊢ f y ∉ q ⟨(t, ⟨s, sb⟩), ⋯⟩
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_1
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement (↑H'') S
⊢ ↑(map H.subtype H'') * ⇑H.subtype '' S = ↑H

pp: case refine_2
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement (↑H'') S
⊢ Nat.card ↥(map H.subtype H'') * Nat.card ↑(⇑H.subtype '' S) = Nat.card ↥H

Tactic:
·

Goal state after:
case refine_1
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement (↑H'') S
⊢ ↑(map H.subtype H'') * ⇑H.subtype '' S = ↑H
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case h₂
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 := ⋯
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := ⋯
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := ⋯
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := ⋯
⊢ ⋂ n, F n ⊆ range f

Tactic:
·

Goal state after:
case h₂
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 := ⋯
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := ⋯
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := ⋯
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := ⋯
⊢ ⋂ n, F n ⊆ range f
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_2
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement (↑H'') S
⊢ Nat.card ↥(map H.subtype H'') * Nat.card ↑(⇑H.subtype '' S) = Nat.card ↥H

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝ : Group G
H' H : Subgroup G
h : H' ≤ H
H'' : Subgroup ↥H := comap H.subtype H'
this : H' = map H.subtype H''
S : Set ↥H
cmem : IsComplement (↑H'') S
⊢ Nat.card ↥(map H.subtype H'') * Nat.card ↑(⇑H.subtype '' S) = Nat.card ↥H
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mp
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

pp: case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ g₁⁻¹ * g₂ ∈ K → (hSK.equiv g₁).1 = (hSK.equiv g₂).1

Tactic:
·

Goal state after:
case mp
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.MeasureTheory.Measure.Regular

Goal state before:
pp: case inl
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
A : Set α
μ : Measure α
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
H : μ A = ∞
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε

pp: case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
A : Set α
μ : Measure α
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
H : μ A ≠ ∞
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
A : Set α
μ : Measure α
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
H : μ A = ∞
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ g₁⁻¹ * g₂ ∈ K → (hSK.equiv g₁).1 = (hSK.equiv g₂).1

Tactic:
·

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

Goal state before:
pp: case mpr.py₁
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₁).1)⁻¹ * g₁ ∈ ↑K

pp: case mpr.py₂
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ ∈ ↑K

Tactic:
·

Goal state after:
case mpr.py₁
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₁).1)⁻¹ * g₁ ∈ ↑K
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
A : Set α
μ : Measure α
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
H : μ A ≠ ∞
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
A : Set α
μ : Measure α
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
H : μ A ≠ ∞
⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr.py₂
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ ∈ ↑K

Tactic:
·

Goal state after:
case mpr.py₂
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ ∈ ↑K
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mp
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 → g₂ * g₁⁻¹ ∈ H

pp: case mpr
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ g₂ * g₁⁻¹ ∈ H → (hHT.equiv g₁).2 = (hHT.equiv g₂).2

Tactic:
·

Goal state after:
case mp
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 → g₂ * g₁⁻¹ ∈ H
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 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
n : ℕ
⊢ (↑(s n)).Nonempty

Tactic:
nonempty_iff_ne_empty

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✝ : 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 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
n : ℕ
⊢ ↑(s n) ≠ ∅
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ g₂ * g₁⁻¹ ∈ H → (hHT.equiv g₁).2 = (hHT.equiv g₂).2

Tactic:
·

Goal state after:
case mpr
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
⊢ g₂ * g₁⁻¹ ∈ H → (hHT.equiv g₁).2 = (hHT.equiv g₂).2
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr.py₁
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₁ * (↑(hHT.equiv g₁).2)⁻¹ ∈ ↑H

pp: case mpr.py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₁ * (↑(hHT.equiv g₂).2)⁻¹ ∈ ↑H

Tactic:
·

Goal state after:
case mpr.py₁
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₁ * (↑(hHT.equiv g₁).2)⁻¹ ∈ ↑H
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr.py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₁ * (↑(hHT.equiv g₂).2)⁻¹ ∈ ↑H

Tactic:
·

Goal state after:
case mpr.py₂
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g₁ g₂ : G
h : g₂ * g₁⁻¹ ∈ H
⊢ g₁ * (↑(hHT.equiv g₂).2)⁻¹ ∈ ↑H
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case refine_3
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 ∈ C, μ t = ν t

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

pp: case refine_5
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
⊢ ∀ (f : ℕ → Set (α × β × γ)),
    Pairwise (Disjoint on f) →
      (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), μ (f i) = ν (f i)) → μ (⋃ i, f i) = ν (⋃ i, f i)

Tactic:
·

Goal state after:
case refine_3
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 ∈ C, μ t = ν t
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case inl
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
hx : 0 ≠ ∞
⊢ (0 • μ).OuterRegular

pp: case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
⊢ (x • μ).OuterRegular

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
hx : 0 ≠ ∞
⊢ (0 • μ).OuterRegular
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
⊢ (x • μ).OuterRegular

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
⊢ (x • μ).OuterRegular
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case fst
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ ↑(hSK.equiv (g * ↑k)).1 = ↑((hSK.equiv g).1, (hSK.equiv g).2 * k).1

pp: case snd
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ ↑(hSK.equiv (g * ↑k)).2 = ↑((hSK.equiv g).1, (hSK.equiv g).2 * k).2

Tactic:
·

Goal state after:
case fst
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ ↑(hSK.equiv (g * ↑k)).1 = ↑((hSK.equiv g).1, (hSK.equiv g).2 * k).1
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case snd
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ ↑(hSK.equiv (g * ↑k)).2 = ↑((hSK.equiv g).1, (hSK.equiv g).2 * k).2

Tactic:
·

Goal state after:
case snd
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ ↑(hSK.equiv (g * ↑k)).2 = ↑((hSK.equiv g).1, (hSK.equiv g).2 * k).2
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_2
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ hHT.equiv (↑h * g) = (h * (hHT.equiv g).1, (hHT.equiv g).2)

pp: case refine_1
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
⊢ RightCosetEquivalence (↑H) (↑h * g) g

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ hHT.equiv (↑h * g) = (h * (hHT.equiv g).1, (hHT.equiv g).2)
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_2.fst
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ ↑(hHT.equiv (↑h * g)).1 = ↑(h * (hHT.equiv g).1, (hHT.equiv g).2).1

pp: case refine_2.snd
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ ↑(hHT.equiv (↑h * g)).2 = ↑(h * (hHT.equiv g).1, (hHT.equiv g).2).2

Tactic:
·

Goal state after:
case refine_2.fst
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ ↑(hHT.equiv (↑h * g)).1 = ↑(h * (hHT.equiv g).1, (hHT.equiv g).2).1
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

pp: case refine_5
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
⊢ ∀ (f : ℕ → Set (α × β × γ)),
    Pairwise (Disjoint on f) →
      (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), μ (f i) = ν (f i)) → μ (⋃ i, f i) = ν (⋃ i, f i)

Tactic:
·

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)), MeasurableSet t → μ t = ν t → μ tᶜ = ν tᶜ