File: Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean

full_name: parallelepiped
code: /-- The closed parallelepiped spanned by a finite family of vectors. -/
def parallelepiped (v : ι → E) : Set E 
goal: Set E

full_name: mem_parallelepiped_iff
code: theorem mem_parallelepiped_iff (v : ι → E) (x : E) :
    x ∈ parallelepiped v ↔ ∃ t ∈ Icc (0 : ι → ℝ) 1, x = ∑ i, t i • v i 
goal: x ∈ parallelepiped v ↔ ∃ t ∈ Set.Icc 0 1, x = ∑ i : ι, t i • v i

full_name: parallelepiped_basis_eq
code: theorem parallelepiped_basis_eq (b : Basis ι ℝ E) :
    parallelepiped b = {x | ∀ i, b.repr x i ∈ Set.Icc 0 1} 
goal: parallelepiped ⇑b = {x | ∀ (i : ι), (b.repr x) i ∈ Set.Icc 0 1}

full_name: image_parallelepiped
code: theorem image_parallelepiped (f : E →ₗ[ℝ] F) (v : ι → E) :
    f '' parallelepiped v = parallelepiped (f ∘ v) 
goal: ⇑f '' parallelepiped v = parallelepiped (⇑f ∘ v)

full_name: parallelepiped_comp_equiv
code: /-- Reindexing a family of vectors does not change their parallelepiped. -/
@[simp]
theorem parallelepiped_comp_equiv (v : ι → E) (e : ι' ≃ ι) :
    parallelepiped (v ∘ e) = parallelepiped v 
goal: parallelepiped (v ∘ ⇑e) = parallelepiped v

full_name: parallelepiped_orthonormalBasis_one_dim
code: theorem parallelepiped_orthonormalBasis_one_dim (b : OrthonormalBasis ι ℝ ℝ) :
    parallelepiped b = Icc 0 1 ∨ parallelepiped b = Icc (-1) 0 
goal: parallelepiped ⇑b = Set.Icc 0 1 ∨ parallelepiped ⇑b = Set.Icc (-1) 0

full_name: parallelepiped_eq_sum_segment
code: theorem parallelepiped_eq_sum_segment (v : ι → E) : parallelepiped v = ∑ i, segment ℝ 0 (v i) 
goal: parallelepiped v = ∑ i : ι, segment ℝ 0 (v i)

full_name: convex_parallelepiped
code: theorem convex_parallelepiped (v : ι → E) : Convex ℝ (parallelepiped v) 
goal: Convex ℝ (parallelepiped v)

full_name: parallelepiped_eq_convexHull
code: /-- A `parallelepiped` is the convex hull of its vertices -/
theorem parallelepiped_eq_convexHull (v : ι → E) :
    parallelepiped v = convexHull ℝ (∑ i, {(0 : E), v i}) 
goal: parallelepiped v = (convexHull ℝ) (∑ i : ι, {0, v i})

full_name: parallelepiped_single
code: /-- The axis aligned parallelepiped over `ι → ℝ` is a cuboid. -/
theorem parallelepiped_single [DecidableEq ι] (a : ι → ℝ) :
    (parallelepiped fun i => Pi.single i (a i)) = Set.uIcc 0 a 
goal: (parallelepiped fun i => Pi.single i (a i)) = Set.uIcc 0 a

full_name: Basis.parallelepiped
code: /-- The parallelepiped spanned by a basis, as a compact set with nonempty interior. -/
def Basis.parallelepiped (b : Basis ι ℝ E) : PositiveCompacts E 
goal: TopologicalSpace.PositiveCompacts E

full_name: Basis.coe_parallelepiped
code: @[simp]
theorem Basis.coe_parallelepiped (b : Basis ι ℝ E) :
    (b.parallelepiped : Set E) = _root_.parallelepiped b 
goal: ↑b.parallelepiped = parallelepiped ⇑b

full_name: Basis.parallelepiped_reindex
code: @[simp]
theorem Basis.parallelepiped_reindex (b : Basis ι ℝ E) (e : ι ≃ ι') :
    (b.reindex e).parallelepiped = b.parallelepiped 
goal: (b.reindex e).parallelepiped = b.parallelepiped

full_name: Basis.parallelepiped_map
code: theorem Basis.parallelepiped_map (b : Basis ι ℝ E) (e : E ≃ₗ[ℝ] F) :
    (b.map e).parallelepiped = b.parallelepiped.map e
    (have := FiniteDimensional.of_fintype_basis b
    -- Porting note: Lean cannot infer the instance above
    LinearMap.continuous_of_finiteDimensional e.toLinearMap)
    (have := FiniteDimensional.of_fintype_basis (b.map e)
    -- Porting note: Lean cannot infer the instance above
    LinearMap.isOpenMap_of_finiteDimensional _ e.surjective) 
goal: (b.map e).parallelepiped = TopologicalSpace.PositiveCompacts.map ⇑e ⋯ ⋯ b.parallelepiped

full_name: Basis.prod_parallelepiped
code: theorem Basis.prod_parallelepiped (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
    (v.prod w).parallelepiped = v.parallelepiped.prod w.parallelepiped 
goal: (v.prod w).parallelepiped = v.parallelepiped.prod w.parallelepiped

full_name: wrapped._@.Mathlib.MeasureTheory.Measure.Haar.OfBasis._hyg.3048
code: /-- The Lebesgue measure associated to a basis, giving measure `1` to the parallelepiped spanned
by the basis. -/
irreducible_def Basis.addHaar (b : Basis ι ℝ E) : Measure E 
goal: Subtype (Eq @definition✝)

full_name: Basis.addHaar_def
code: /-- The Lebesgue measure associated to a basis, giving measure `1` to the parallelepiped spanned
by the basis. -/
irreducible_def Basis.addHaar (b : Basis ι ℝ E) : Measure E 
goal: b.addHaar = MeasureTheory.Measure.addHaarMeasure b.parallelepiped

full_name: definition._@.Mathlib.MeasureTheory.Measure.Haar.OfBasis._hyg.3048
code: /-- The Lebesgue measure associated to a basis, giving measure `1` to the parallelepiped spanned
by the basis. -/
irreducible_def Basis.addHaar (b : Basis ι ℝ E) : Measure E 
goal: MeasureTheory.Measure E

full_name: Basis.addHaar
code: /-- The Lebesgue measure associated to a basis, giving measure `1` to the parallelepiped spanned
by the basis. -/
irreducible_def Basis.addHaar (b : Basis ι ℝ E) : Measure E 
goal: MeasureTheory.Measure E

full_name: IsAddHaarMeasure_basis_addHaar
code: instance IsAddHaarMeasure_basis_addHaar (b : Basis ι ℝ E) : IsAddHaarMeasure b.addHaar 
goal: b.addHaar.IsAddHaarMeasure

full_name: instSigmaFiniteAddHaar
code: instance (b : Basis ι ℝ E) : SigmaFinite b.addHaar 
goal: MeasureTheory.SigmaFinite b.addHaar

full_name: Basis.addHaar_eq_iff
code: /-- Let `μ` be a σ-finite left invariant measure on `E`. Then `μ` is equal to the Haar measure
defined by `b` iff the parallelepiped defined by `b` has measure `1` for `μ`. -/
theorem Basis.addHaar_eq_iff [SecondCountableTopology E] (b : Basis ι ℝ E) (μ : Measure E)
    [SigmaFinite μ] [IsAddLeftInvariant μ] :
    b.addHaar = μ ↔ μ b.parallelepiped = 1 
goal: b.addHaar = μ ↔ μ ↑b.parallelepiped = 1

full_name: Basis.addHaar_reindex
code: @[simp]
theorem Basis.addHaar_reindex (b : Basis ι ℝ E) (e : ι ≃ ι') :
    (b.reindex e).addHaar = b.addHaar 
goal: (b.reindex e).addHaar = b.addHaar

full_name: Basis.addHaar_self
code: theorem Basis.addHaar_self (b : Basis ι ℝ E) : b.addHaar (_root_.parallelepiped b) = 1 
goal: b.addHaar (parallelepiped ⇑b) = 1

full_name: Basis.prod_addHaar
code: theorem Basis.prod_addHaar (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
    (v.prod w).addHaar = v.addHaar.prod w.addHaar 
goal: (v.prod w).addHaar = v.addHaar.prod w.addHaar

full_name: measureSpaceOfInnerProductSpace
code: /-- A finite dimensional inner product space has a canonical measure, the Lebesgue measure giving
volume `1` to the parallelepiped spanned by any orthonormal basis. We define the measure using
some arbitrary choice of orthonormal basis. The fact that it works with any orthonormal basis
is proved in `orthonormalBasis.volume_parallelepiped`. -/
instance (priority := 100) measureSpaceOfInnerProductSpace [NormedAddCommGroup E]
    [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] :
    MeasureSpace E 
goal: MeasureTheory.MeasureSpace E

full_name: instIsAddHaarMeasureVolume
code: instance [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
    [MeasurableSpace E] [BorelSpace E] : IsAddHaarMeasure (volume : Measure E) 
goal: MeasureTheory.volume.IsAddHaarMeasure

full_name: Real.measureSpace
code: instance Real.measureSpace : MeasureSpace ℝ 
goal: MeasureTheory.MeasureSpace ℝ

full_name: EuclideanSpace.instMeasurableSpaceReal
code: instance : MeasurableSpace (EuclideanSpace ℝ ι) 
goal: MeasurableSpace (EuclideanSpace ℝ ι)

full_name: EuclideanSpace.instBorelSpaceRealOfFinite
code: instance [Finite ι] : BorelSpace (EuclideanSpace ℝ ι) 
goal: BorelSpace (EuclideanSpace ℝ ι)

full_name: EuclideanSpace.measurableEquiv
code: /-- `WithLp.equiv` as a `MeasurableEquiv`. -/
@[simps toEquiv]
protected def measurableEquiv : EuclideanSpace ℝ ι ≃ᵐ (ι → ℝ) 
goal: EuclideanSpace ℝ ι ≃ᵐ (ι → ℝ)

full_name: EuclideanSpace.measurableEquiv_toEquiv
code: /-- `WithLp.equiv` as a `MeasurableEquiv`. -/
@[simps toEquiv]
protected def measurableEquiv : EuclideanSpace ℝ ι ≃ᵐ (ι → ℝ) 
goal: (EuclideanSpace.measurableEquiv ι).toEquiv = WithLp.equiv 2 ((i : ι) → (fun x => ℝ) i)

full_name: EuclideanSpace.coe_measurableEquiv
code: theorem coe_measurableEquiv : ⇑(EuclideanSpace.measurableEquiv ι) = WithLp.equiv 2 _ 
goal: ⇑(EuclideanSpace.measurableEquiv ι) = ⇑(WithLp.equiv 2 ((i : ι) → (fun x => ℝ) i))

full_name: EuclideanSpace.coe_measurableEquiv_symm
code: theorem coe_measurableEquiv_symm :
    ⇑(EuclideanSpace.measurableEquiv ι).symm = (WithLp.equiv 2 _).symm 
goal: ⇑(EuclideanSpace.measurableEquiv ι).symm = ⇑(WithLp.equiv 2 (ι → ℝ)).symm