mathlib-initiative/mathlib-tactics huggingface dataset:
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
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ hSK.equiv (g * ↑k) = ((hSK.equiv g).1, (hSK.equiv g).2 * k)

Tactic:
ext

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

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)

Tactic:
ext

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

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
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ T
⊢ ↑(hST.equiv g).1 = g ↔ g ∈ S

Tactic:
constructor

Goal state after:
case mp
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ T
⊢ ↑(hST.equiv g).1 = g → g ∈ S

case mpr
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ T
⊢ g ∈ S → ↑(hST.equiv g).1 = g
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
⊢ ↑(hST.equiv g).2 = g ↔ g ∈ T

Tactic:
constructor

Goal state after:
case mp
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
⊢ ↑(hST.equiv g).2 = g → g ∈ T

case mpr
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
⊢ g ∈ T → ↑(hST.equiv g).2 = g
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: 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
⊢ t' = ⋃ c ∈ u, t ∩ g ⁻¹' {c}

Tactic:
ext

Goal state after:
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
x✝ : α
⊢ x✝ ∈ t' ↔ x✝ ∈ ⋃ c ∈ u, t ∩ g ⁻¹' {c}
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_4
β : Type u_5
γ : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝¹ : SFinite μ
inst✝ : SFinite ν
h2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving (fun x ↦ f (x, y)) μ τ
⊢ QuasiMeasurePreserving f (map Prod.swap (ν.prod μ)) τ

Tactic:
convert!
    (QuasiMeasurePreserving.prod_of_right (hf.comp measurable_swap) h2f).comp
      ((measurable_swap.measurePreserving (ν.prod μ)).symm
          MeasurableEquiv.prodComm).quasiMeasurePreserving

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

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulLeftInvariant
inst✝ : SFinite ν
⊢ (μ.prod ν).IsMulLeftInvariant

Tactic:
constructor

Goal state after:
G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulLeftInvariant
inst✝ : SFinite ν
⊢ ∀ (g : G × H), map (fun x ↦ g * x) (μ.prod ν) = μ.prod ν
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulRightInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SFinite ν
⊢ (μ.prod ν).IsMulRightInvariant

Tactic:
constructor

Goal state after:
G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulRightInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SFinite ν
⊢ ∀ (g : G × H), map (fun x ↦ x * g) (μ.prod ν) = μ.prod ν
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹¹ : TopologicalSpace α
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : BorelSpace α
inst✝⁸ : LinearOrder α
inst✝⁷ : OrderTopology α
inst✝⁶ : PolishSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : LinearOrder β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
t' : Set α := {x | x ∈ t ∧ ¬ContinuousWithinAt g t x}
ht' : t'.Countable
⊢ t = t \ t' ∪ t'

Tactic:
ext

Goal state after:
α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹¹ : TopologicalSpace α
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : BorelSpace α
inst✝⁸ : LinearOrder α
inst✝⁷ : OrderTopology α
inst✝⁶ : PolishSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : LinearOrder β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
t' : Set α := {x | x ∈ t ∧ ¬ContinuousWithinAt g t x}
ht' : t'.Countable
x✝ : α
⊢ x✝ ∈ t ↔ x✝ ∈ t \ t' ∪ t'
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
⊢ μ.inv.IsMulRightInvariant

Tactic:
constructor

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
⊢ μ.inv.IsMulLeftInvariant

Tactic:
constructor

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.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
⊢ ⇑f '' (fun x ↦ g * x) ⁻¹' s = (fun x ↦ f g * x) ⁻¹' ⇑f '' s

Tactic:
ext

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ g * x) ⁻¹' s ↔ x✝ ∈ (fun x ↦ f g * x) ⁻¹' ⇑f '' s
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ g * x) ⁻¹' s ↔ x✝ ∈ (fun x ↦ f g * x) ⁻¹' ⇑f '' s

Tactic:
constructor

Goal state after:
case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ g * x) ⁻¹' s → x✝ ∈ (fun x ↦ f g * x) ⁻¹' ⇑f '' s

case mpr
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ (fun x ↦ f g * x) ⁻¹' ⇑f '' s → x✝ ∈ ⇑f '' (fun x ↦ g * x) ⁻¹' s
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
⊢ ⇑f '' (fun x ↦ x * g) ⁻¹' s = (fun x ↦ x * f g) ⁻¹' ⇑f '' s

Tactic:
ext

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s ↔ x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s ↔ x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s

Tactic:
constructor

Goal state after:
case mp
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s → x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s

case mpr
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
x✝ : H
⊢ x✝ ∈ (fun x ↦ x * f g) ⁻¹' ⇑f '' s → x✝ ∈ ⇑f '' (fun x ↦ x * g) ⁻¹' s
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : StandardBorelSpace α
s : Set α
hs : MeasurableSet s
this : UpgradedStandardBorel α := upgradeStandardBorel α
t : TopologicalSpace α
hle : t ≤ this.toTopologicalSpace
ht : PolishSpace α
s_clopen : IsClosed[t] s ∧ IsOpen[t] s
⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s

Tactic:
refine ⟨t, ?_, ht, s_clopen⟩

Goal state after:
α : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : StandardBorelSpace α
s : Set α
hs : MeasurableSet s
this : UpgradedStandardBorel α := upgradeStandardBorel α
t : TopologicalSpace α
hle : t ≤ this.toTopologicalSpace
ht : PolishSpace α
s_clopen : IsClosed[t] s ∧ IsOpen[t] s
⊢ BorelSpace α
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : StandardBorelSpace α
s : Set α
hs : MeasurableSet s
this : UpgradedStandardBorel α := upgradeStandardBorel α
t : TopologicalSpace α
hle : t ≤ this.toTopologicalSpace
ht : PolishSpace α
s_clopen : IsClosed[t] s ∧ IsOpen[t] s
⊢ BorelSpace α

Tactic:
constructor

Goal state after:
α : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : StandardBorelSpace α
s : Set α
hs : MeasurableSet s
this : UpgradedStandardBorel α := ⋯
t : TopologicalSpace α
hle : t ≤ this.toTopologicalSpace
ht : PolishSpace α
s_clopen : IsClosed[t] s ∧ IsOpen[t] s
⊢ inst✝¹ = borel α
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:
by
  rw [Measure.snd, Measure.map_map measurable_snd measurable_swap]
  rfl

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace α
μ ν : Measure α
inst✝¹ : μ.InnerRegular
inst✝ : ν.InnerRegular
⊢ (μ + ν).InnerRegular

Tactic:
constructor

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace α
μ ν : Measure α
inst✝¹ : μ.InnerRegular
inst✝ : ν.InnerRegular
⊢ (μ + ν).InnerRegularWRT IsCompact MeasurableSet
Module: Mathlib.Analysis.LConvolution

Goal state before:
pp: G : Type u_1
mG : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : Inv G
f : G → ℝ≥0∞
μ : Measure G
⊢ 0 ⋆ₘₗ[μ] f = 0

Tactic:
ext

Goal state after:
G : Type u_1
mG : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : Inv G
f : G → ℝ≥0∞
μ : Measure G
x✝ : G
⊢ (0 ⋆ₘₗ[μ] f) x✝ = 0 x✝
Module: Mathlib.Analysis.LConvolution

Goal state before:
pp: G : Type u_1
mG : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : Inv G
f : G → ℝ≥0∞
μ : Measure G
⊢ f ⋆ₘₗ[μ] 0 = 0

Tactic:
ext

Goal state after:
G : Type u_1
mG : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : Inv G
f : G → ℝ≥0∞
μ : Measure G
x✝ : G
⊢ (f ⋆ₘₗ[μ] 0) x✝ = 0 x✝
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ✝ : Measure α
inst✝¹ : TopologicalSpace α
ι : Type u_3
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).InnerRegular
⊢ (sum μ).InnerRegular

Tactic:
constructor

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ✝ : Measure α
inst✝¹ : TopologicalSpace α
ι : Type u_3
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).InnerRegular
⊢ (sum μ).InnerRegularWRT IsCompact MeasurableSet
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
inst✝ : IsFiniteMeasure μ
⊢ μ.InnerRegular

Tactic:
constructor

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
inst✝ : IsFiniteMeasure μ
⊢ μ.InnerRegularWRT IsCompact MeasurableSet
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ (μ.withDensity f) s = 0 ↔ μ ({x | f x ≠ 0} ∩ s) = 0

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ (μ.withDensity f) s = 0 → μ ({x | f x ≠ 0} ∩ s) = 0

case mpr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ μ ({x | f x ≠ 0} ∩ s) = 0 → (μ.withDensity f) s = 0
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hs : NullMeasurableSet s μ
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ U, IsOpen[inst✝⁴] U ∧ μ U < ∞ ∧ μ (U ∆ s) < ε

Tactic:
by
  rcases hs with ⟨t, htm, hst⟩
  rcases htm.exists_isOpen_symmDiff_lt (by rwa [← measure_congr hst]) hε with ⟨U, hUo, hμU, hUs⟩
  refine ⟨U, hUo, hμU, ?_⟩
  rwa [measure_congr <| (ae_eq_refl _).symmDiff hst]

Goal state after:
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:
constructor

Goal state after:
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
⊢ ∀ (x : G), μ {x} = 0
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ∞
⊢ (c • μ).InnerRegularCompactLTTop

Tactic:
constructor

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : c = ∞
⊢ (c • μ).InnerRegularWRT IsCompact fun s ↦ MeasurableSet s ∧ (c • μ) s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : ¬c = ∞
⊢ (c • μ).InnerRegularCompactLTTop

Tactic:
constructor

Goal state after:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace α
h : μ.InnerRegularCompactLTTop
c : ℝ≥0∞
hc : ¬c = 0
h'c : ¬c = ∞
⊢ (c • μ).InnerRegularWRT IsCompact fun s ↦ MeasurableSet s ∧ (c • μ) s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous[inst✝⁴, inst✝¹] f
⊢ (map f μ).InnerRegularCompactLTTop

Tactic:
constructor

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous[inst✝⁴, inst✝¹] f
⊢ (map f μ).InnerRegularWRT IsCompact fun s ↦ MeasurableSet s ∧ (map f μ) s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ∞
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).Regular

Tactic:
constructor

Goal state after:
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ∞
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).InnerRegularWRT IsCompact IsOpen[inst✝³]
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
⊢ AEMeasurable g (μ.withDensity fun x ↦ ↑(f x)) ↔ AEMeasurable (fun x ↦ ↑(f x) * g x) μ

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
⊢ AEMeasurable g (μ.withDensity fun x ↦ ↑(f x)) → AEMeasurable (fun x ↦ ↑(f x) * g x) μ

case mpr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
⊢ AEMeasurable (fun x ↦ ↑(f x) * g x) μ → AEMeasurable g (μ.withDensity fun x ↦ ↑(f x))
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
⊢ √x = y ↔ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0

Tactic:
constructor

Goal state after:
case mp
x y : ℝ
⊢ √x = y → y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0

case mpr
x y : ℝ
⊢ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0 → √x = y
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
h : 0 ≤ y
⊢ x ^ 2 ≤ y ↔ -√y ≤ x ∧ x ≤ √y

Tactic:
constructor

Goal state after:
case mp
x y : ℝ
h : 0 ≤ y
⊢ x ^ 2 ≤ y → -√y ≤ x ∧ x ≤ √y

case mpr
x y : ℝ
h : 0 ≤ y
⊢ -√y ≤ x ∧ x ≤ √y → x ^ 2 ≤ y
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ √x / x = (√x)⁻¹

Tactic:
rw [sqrt_div_self', one_div]

Goal state after:
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: r : ℝ
⊢ ‖↑r‖₊ = ‖r‖₊

Tactic:
ext

Goal state after:
r : ℝ
⊢ ↑‖↑r‖₊ = ↑‖r‖₊
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f' = ∫⁻ (a : α), (f' * g) a ∂μ

Tactic:
apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := ⋯
⊢ ∀ᵐ (x : α) ∂μ, AEMeasurable.mk f hf x < ∞
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: p : ℝ × ℝ
⊢ equivRealProd.symm p = ↑p.1 + ↑p.2 * I

Tactic:
ext

Goal state after:
case a
p : ℝ × ℝ
⊢ (equivRealProd.symm p).re = (↑p.1 + ↑p.2 * I).re

case a
p : ℝ × ℝ
⊢ (equivRealProd.symm p).im = (↑p.1 + ↑p.2 * I).im
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: a✝ b✝ c✝ : ℂ
⊢ a✝ + b✝ + c✝ = a✝ + (b✝ + c✝)

Tactic:
ext

Goal state after:
case a
a✝ b✝ c✝ : ℂ
⊢ (a✝ + b✝ + c✝).re = (a✝ + (b✝ + c✝)).re

case a
a✝ b✝ c✝ : ℂ
⊢ (a✝ + b✝ + c✝).im = (a✝ + (b✝ + c✝)).im
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: a✝ : ℂ
⊢ 0 + a✝ = a✝

Tactic:
ext

Goal state after:
case a
a✝ : ℂ
⊢ (0 + a✝).re = a✝.re

case a
a✝ : ℂ
⊢ (0 + a✝).im = a✝.im
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: a✝ : ℂ
⊢ a✝ + 0 = a✝

Tactic:
ext

Goal state after:
case a
a✝ : ℂ
⊢ (a✝ + 0).re = a✝.re

case a
a✝ : ℂ
⊢ (a✝ + 0).im = a✝.im
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: x✝ : ℂ
⊢ 0 • x✝ = 0

Tactic:
ext

Goal state after:
case a
x✝ : ℂ
⊢ (0 • x✝).re = re 0

case a
x✝ : ℂ
⊢ (0 • x✝).im = im 0
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: n✝ : ℕ
x✝ : ℂ
⊢ (n✝ + 1) • x✝ = n✝ • x✝ + x✝

Tactic:
ext

Goal state after:
case a
n✝ : ℕ
x✝ : ℂ
⊢ ((n✝ + 1) • x✝).re = (n✝ • x✝ + x✝).re

case a
n✝ : ℕ
x✝ : ℂ
⊢ ((n✝ + 1) • x✝).im = (n✝ • x✝ + x✝).im
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: a✝ : ℂ
⊢ 0 • a✝ = 0

Tactic:
ext

Goal state after:
case a
a✝ : ℂ
⊢ (0 • a✝).re = re 0

case a
a✝ : ℂ
⊢ (0 • a✝).im = im 0