ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Zspan.span_top
theorem span_top : span K (span ℤ (Set.range b) : Set E) = ⊤ 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E)
Goal: Submodule.span K ↑(Submodule.span ℤ (Set.range ⇑b)) = ⊤
-- Zspan.mem_fundamentalDomain
@[simp]
theorem mem_fundamentalDomain {m : E} :
    m ∈ fundamentalDomain b ↔ ∀ i, b.repr m i ∈ Set.Ico (0 : K) 1 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) {m : E}
Goal: m ∈ Zspan.fundamentalDomain b ↔ ∀ (i : ι), (b.repr m) i ∈ Set.Ico 0 1
-- Zspan.map_fundamentalDomain
theorem map_fundamentalDomain {F : Type*} [NormedAddCommGroup F] [NormedSpace K F] (f : E ≃ₗ[K] F) :
    f '' (fundamentalDomain b) = fundamentalDomain (b.map f) 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) {F : Type u_4} [NormedAddCommGroup F] [NormedSpace K F] (f : E ≃ₗ[K] F)
Goal: ⇑f '' Zspan.fundamentalDomain b = Zspan.fundamentalDomain (b.map f)
-- Zspan.fundamentalDomain_reindex
@[simp]
theorem fundamentalDomain_reindex {ι' : Type*} (e : ι ≃ ι') :
    fundamentalDomain (b.reindex e) = fundamentalDomain b 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) {ι' : Type u_4} (e : ι ≃ ι')
Goal: Zspan.fundamentalDomain (b.reindex e) = Zspan.fundamentalDomain b
-- Zspan.fundamentalDomain_pi_basisFun
lemma fundamentalDomain_pi_basisFun [Fintype ι] :
    fundamentalDomain (Pi.basisFun ℝ ι) = Set.pi Set.univ fun _ : ι ↦ Set.Ico (0 : ℝ) 1 
Arguments: {ι : Type u_2} [Fintype ι]
Goal: Zspan.fundamentalDomain (Pi.basisFun ℝ ι) = Set.univ.pi fun x => Set.Ico 0 1
-- Zspan.repr_floor_apply
@[simp]
theorem repr_floor_apply (m : E) (i : ι) : b.repr (floor b m) i = ⌊b.repr m i⌋ 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (i : ι)
Goal: (b.repr ↑(Zspan.floor b m)) i = ↑⌊(b.repr m) i⌋
-- Zspan.repr_ceil_apply
@[simp]
theorem repr_ceil_apply (m : E) (i : ι) : b.repr (ceil b m) i = ⌈b.repr m i⌉ 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (i : ι)
Goal: (b.repr ↑(Zspan.ceil b m)) i = ↑⌈(b.repr m) i⌉
-- Zspan.floor_eq_self_of_mem
@[simp]
theorem floor_eq_self_of_mem (m : E) (h : m ∈ span ℤ (Set.range b)) : (floor b m : E) = m 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (h : m ∈ Submodule.span ℤ (Set.range ⇑b))
Goal: ↑(Zspan.floor b m) = m
-- Zspan.ceil_eq_self_of_mem
@[simp]
theorem ceil_eq_self_of_mem (m : E) (h : m ∈ span ℤ (Set.range b)) : (ceil b m : E) = m 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (h : m ∈ Submodule.span ℤ (Set.range ⇑b))
Goal: ↑(Zspan.ceil b m) = m
-- Zspan.fract_apply
theorem fract_apply (m : E) : fract b m = m - floor b m 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E)
Goal: Zspan.fract b m = m - ↑(Zspan.floor b m)
-- Zspan.repr_fract_apply
@[simp]
theorem repr_fract_apply (m : E) (i : ι) : b.repr (fract b m) i = Int.fract (b.repr m i) 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (i : ι)
Goal: (b.repr (Zspan.fract b m)) i = Int.fract ((b.repr m) i)
-- Zspan.fract_fract
@[simp]
theorem fract_fract (m : E) : fract b (fract b m) = fract b m 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E)
Goal: Zspan.fract b (Zspan.fract b m) = Zspan.fract b m
-- Zspan.fract_zspan_add
@[simp]
theorem fract_zspan_add (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
    fract b (v + m) = fract b m 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) {v : E} (h : v ∈ Submodule.span ℤ (Set.range ⇑b))
Goal: Zspan.fract b (v + m) = Zspan.fract b m
-- Zspan.fract_add_zspan
@[simp]
theorem fract_add_zspan (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
    fract b (m + v) = fract b m 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) {v : E} (h : v ∈ Submodule.span ℤ (Set.range ⇑b))
Goal: Zspan.fract b (m + v) = Zspan.fract b m
-- Zspan.fract_eq_self
theorem fract_eq_self {x : E} : fract b x = x ↔ x ∈ fundamentalDomain b 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] {b : Basis ι K E} [FloorRing K] [Fintype ι] {x : E}
Goal: Zspan.fract b x = x ↔ x ∈ Zspan.fundamentalDomain b
-- Zspan.fract_mem_fundamentalDomain
theorem fract_mem_fundamentalDomain (x : E) : fract b x ∈ fundamentalDomain b 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E)
Goal: Zspan.fract b x ∈ Zspan.fundamentalDomain b
-- Zspan.fractRestrict_surjective
theorem fractRestrict_surjective : Function.Surjective (fractRestrict b) 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι]
Goal: Function.Surjective (Zspan.fractRestrict b)
-- Zspan.fractRestrict_apply
@[simp]
theorem fractRestrict_apply (x : E) : (fractRestrict b x : E) = fract b x 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E)
Goal: ↑(Zspan.fractRestrict b x) = Zspan.fract b x
-- Zspan.fract_eq_fract
theorem fract_eq_fract (m n : E) : fract b m = fract b n ↔ -m + n ∈ span ℤ (Set.range b) 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m n : E)
Goal: Zspan.fract b m = Zspan.fract b n ↔ -m + n ∈ Submodule.span ℤ (Set.range ⇑b)
-- Zspan.norm_fract_le
theorem norm_fract_le [HasSolidNorm K] (m : E) : ‖fract b m‖ ≤ ∑ i, ‖b i‖ 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] [HasSolidNorm K] (m : E)
Goal: ‖Zspan.fract b m‖ ≤ ∑ i : ι, ‖b i‖
-- Zspan.coe_floor_self
@[simp]
theorem coe_floor_self (k : K) : (floor (Basis.singleton ι K) k : K) = ⌊k⌋ 
Arguments: {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [FloorRing K] [Fintype ι] [Unique ι] (k : K)
Goal: ↑(Zspan.floor (Basis.singleton ι K) k) = ↑⌊k⌋
-- Zspan.coe_fract_self
@[simp]
theorem coe_fract_self (k : K) : (fract (Basis.singleton ι K) k : K) = Int.fract k 
Arguments: {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [FloorRing K] [Fintype ι] [Unique ι] (k : K)
Goal: Zspan.fract (Basis.singleton ι K) k = Int.fract k
-- Zspan.fundamentalDomain_isBounded
theorem fundamentalDomain_isBounded [Finite ι] [HasSolidNorm K] :
    IsBounded (fundamentalDomain b) 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Finite ι] [HasSolidNorm K]
Goal: Bornology.IsBounded (Zspan.fundamentalDomain b)
-- Zspan.vadd_mem_fundamentalDomain
theorem vadd_mem_fundamentalDomain [Fintype ι] (y : span ℤ (Set.range b)) (x : E) :
    y +ᵥ x ∈ fundamentalDomain b ↔ y = -floor b x 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (y : ↥(Submodule.span ℤ (Set.range ⇑b))) (x : E)
Goal: y +ᵥ x ∈ Zspan.fundamentalDomain b ↔ y = -Zspan.floor b x
-- Zspan.exist_unique_vadd_mem_fundamentalDomain
theorem exist_unique_vadd_mem_fundamentalDomain [Finite ι] (x : E) :
    ∃! v : span ℤ (Set.range b), v +ᵥ x ∈ fundamentalDomain b 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Finite ι] (x : E)
Goal: ∃! v, v +ᵥ x ∈ Zspan.fundamentalDomain b
-- Zspan.quotientEquiv_apply_mk
@[simp]
theorem quotientEquiv_apply_mk [Fintype ι] (x : E) :
    quotientEquiv b (Submodule.Quotient.mk x) = fractRestrict b x 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E)
Goal: (Zspan.quotientEquiv b) (Submodule.Quotient.mk x) = Zspan.fractRestrict b x
-- Zspan.quotientEquiv.symm_apply
@[simp]
theorem quotientEquiv.symm_apply [Fintype ι] (x : fundamentalDomain b) :
    (quotientEquiv b).symm x = Submodule.Quotient.mk ↑x 
Arguments: {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : ↑(Zspan.fundamentalDomain b))
Goal: (Zspan.quotientEquiv b).symm x = Submodule.Quotient.mk ↑x
-- Zspan.discreteTopology_pi_basisFun
theorem discreteTopology_pi_basisFun [Finite ι] :
    DiscreteTopology (span ℤ (Set.range (Pi.basisFun ℝ ι))) 
Arguments: {ι : Type u_2} [Finite ι]
Goal: DiscreteTopology ↥(Submodule.span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
-- Zspan.fundamentalDomain_subset_parallelepiped
theorem fundamentalDomain_subset_parallelepiped [Fintype ι] :
    fundamentalDomain b ⊆ parallelepiped b 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Fintype ι]
Goal: Zspan.fundamentalDomain b ⊆ parallelepiped ⇑b
-- Zspan.fundamentalDomain_measurableSet
@[measurability]
theorem fundamentalDomain_measurableSet [MeasurableSpace E] [OpensMeasurableSpace E] [Finite ι] :
    MeasurableSet (fundamentalDomain b) 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [MeasurableSpace E] [OpensMeasurableSpace E] [Finite ι]
Goal: MeasurableSet (Zspan.fundamentalDomain b)
-- Zspan.isAddFundamentalDomain
protected theorem isAddFundamentalDomain [Finite ι] [MeasurableSpace E] [OpensMeasurableSpace E]
    (μ : Measure E) :
    IsAddFundamentalDomain (span ℤ (Set.range b)).toAddSubgroup (fundamentalDomain b) μ 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Finite ι] [MeasurableSpace E] [OpensMeasurableSpace E] (μ : MeasureTheory.Measure E)
Goal: MeasureTheory.IsAddFundamentalDomain (↥(Submodule.span ℤ (Set.range ⇑b)).toAddSubgroup) (Zspan.fundamentalDomain b) μ
-- Zspan.measure_fundamentalDomain_ne_zero
theorem measure_fundamentalDomain_ne_zero [Finite ι] [MeasurableSpace E] [BorelSpace E]
    {μ : Measure E} [Measure.IsAddHaarMeasure μ] :
    μ (fundamentalDomain b) ≠ 0 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Finite ι] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure]
Goal: μ (Zspan.fundamentalDomain b) ≠ 0
-- Zspan.measure_fundamentalDomain
theorem measure_fundamentalDomain [Fintype ι] [DecidableEq ι] [MeasurableSpace E] (μ : Measure E)
    [BorelSpace E] [Measure.IsAddHaarMeasure μ] (b₀ : Basis ι ℝ E) :
    μ (fundamentalDomain b) = ENNReal.ofReal |b₀.det b| * μ (fundamentalDomain b₀) 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Fintype ι] [DecidableEq ι] [MeasurableSpace E] (μ : MeasureTheory.Measure E) [BorelSpace E] [μ.IsAddHaarMeasure] (b₀ : Basis ι ℝ E)
Goal: μ (Zspan.fundamentalDomain b) = ENNReal.ofReal |b₀.det ⇑b| * μ (Zspan.fundamentalDomain b₀)
-- Zspan.volume_fundamentalDomain
@[simp]
theorem volume_fundamentalDomain [Fintype ι] [DecidableEq ι] (b : Basis ι ℝ (ι → ℝ)) :
    volume (fundamentalDomain b) = ENNReal.ofReal |(Matrix.of b).det| 
Arguments: {ι : Type u_2} [Fintype ι] [DecidableEq ι] (b : Basis ι ℝ (ι → ℝ))
Goal: MeasureTheory.volume (Zspan.fundamentalDomain b) = ENNReal.ofReal |(Matrix.of ⇑b).det|
-- Zspan.fundamentalDomain_ae_parallelepiped
theorem fundamentalDomain_ae_parallelepiped [Fintype ι] [MeasurableSpace E] (μ : Measure E)
    [BorelSpace E] [Measure.IsAddHaarMeasure μ] :
    fundamentalDomain b =ᵐ[μ] parallelepiped b 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Fintype ι] [MeasurableSpace E] (μ : MeasureTheory.Measure E) [BorelSpace E] [μ.IsAddHaarMeasure]
Goal: Zspan.fundamentalDomain b =ᵐ[μ] parallelepiped ⇑b
-- IsZlattice.span_top
class IsZlattice (K : Type*) [NormedField K] {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]
    (L : AddSubgroup E) [DiscreteTopology L] : Prop where
    span_top : span K (L : Set E) = ⊤
Arguments: {K : Type u_1} [NormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {L : AddSubgroup E} [DiscreteTopology ↥L] [self : IsZlattice K L]
Goal: Submodule.span K ↑L = ⊤
-- Zspan.isZlattice
theorem _root_.Zspan.isZlattice {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [Finite ι] (b : Basis ι ℝ E) :
    IsZlattice ℝ (span ℤ (Set.range b)).toAddSubgroup 
Arguments: {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [Finite ι] (b : Basis ι ℝ E)
Goal: IsZlattice ℝ (Submodule.span ℤ (Set.range ⇑b)).toAddSubgroup
-- Zlattice.FG
theorem Zlattice.FG [hs : IsZlattice K L] : AddSubgroup.FG L 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [hs : IsZlattice K L]
Goal: L.FG
-- Zlattice.module_finite
theorem Zlattice.module_finite [IsZlattice K L] : Module.Finite ℤ L 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [IsZlattice K L]
Goal: Module.Finite ℤ ↥L
-- Zlattice.module_free
theorem Zlattice.module_free [IsZlattice K L] : Module.Free ℤ L 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [IsZlattice K L]
Goal: Module.Free ℤ ↥L
-- Zlattice.rank
theorem Zlattice.rank [hs : IsZlattice K L] : finrank ℤ L = finrank K E 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [hs : IsZlattice K L]
Goal: FiniteDimensional.finrank ℤ ↥L = FiniteDimensional.finrank K E
-- Basis.ofZlatticeBasis_apply
@[simp]
theorem Basis.ofZlatticeBasis_apply (i : ι) :
    b.ofZlatticeBasis K L i =  b i 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] {ι : Type u_3} [hs : IsZlattice K L] (b : Basis ι ℤ ↥L) (i : ι)
Goal: (Basis.ofZlatticeBasis K L b) i = ↑(b i)
-- Basis.ofZlatticeBasis_repr_apply
@[simp]
theorem Basis.ofZlatticeBasis_repr_apply (x : L) (i : ι) :
    (b.ofZlatticeBasis K L).repr x i = b.repr x i 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] {ι : Type u_3} [hs : IsZlattice K L] (b : Basis ι ℤ ↥L) (x : ↥L) (i : ι)
Goal: ((Basis.ofZlatticeBasis K L b).repr ↑x) i = ↑((b.repr x) i)
-- Basis.ofZlatticeBasis_span
theorem Basis.ofZlatticeBasis_span :
    (span ℤ (Set.range (b.ofZlatticeBasis K))).toAddSubgroup = L 
Arguments: (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] {ι : Type u_3} [hs : IsZlattice K L] (b : Basis ι ℤ ↥L)
Goal: (Submodule.span ℤ (Set.range ⇑(Basis.ofZlatticeBasis K L b))).toAddSubgroup = L
-- Zlattice.isAddFundamentalDomain
theorem Zlattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] {L : AddSubgroup E} [DiscreteTopology L] [IsZlattice ℝ L] [Finite ι]
    (b : Basis ι ℤ L) [MeasurableSpace E] [OpensMeasurableSpace E] (μ : MeasureTheory.Measure E) :
    MeasureTheory.IsAddFundamentalDomain L (Zspan.fundamentalDomain (b.ofZlatticeBasis ℝ)) μ 
Arguments: {ι : Type u_3} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {L : AddSubgroup E} [DiscreteTopology ↥L] [IsZlattice ℝ L] [Finite ι] (b : Basis ι ℤ ↥L) [MeasurableSpace E] [OpensMeasurableSpace E] (μ : MeasureTheory.Measure E)
Goal: MeasureTheory.IsAddFundamentalDomain (↥L) (Zspan.fundamentalDomain (Basis.ofZlatticeBasis ℝ L b)) μ
-- LieAlgebra.IsKilling.ker_restrict_eq_bot_of_isCartanSubalgebra
lemma ker_restrict_eq_bot_of_isCartanSubalgebra
    [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
    LinearMap.ker ((killingForm R L).restrict H) = ⊥ 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsKilling R L] [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
Goal: LinearMap.ker ((killingForm R L).restrict H.toSubmodule) = ⊥
-- LieAlgebra.IsKilling.restrict_killingForm
lemma restrict_killingForm (H : LieSubalgebra R L) :
    (killingForm R L).restrict H = LieModule.traceForm R H L 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L)
Goal: (killingForm R L).restrict H.toSubmodule = LieModule.traceForm R (↥H) L
-- LieAlgebra.IsKilling.ker_traceForm_eq_bot_of_isCartanSubalgebra
@[simp] lemma ker_traceForm_eq_bot_of_isCartanSubalgebra
    [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
    LinearMap.ker (LieModule.traceForm R H L) = ⊥ 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsKilling R L] [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
Goal: LinearMap.ker (LieModule.traceForm R (↥H) L) = ⊥
-- LieAlgebra.IsKilling.traceForm_cartan_nondegenerate
lemma traceForm_cartan_nondegenerate
    [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
    (LieModule.traceForm R H L).Nondegenerate 
Arguments: (R : Type u_1) (L : Type u_3) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsKilling R L] [IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
Goal: (LieModule.traceForm R (↥H) L).Nondegenerate
-- LieAlgebra.killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero
lemma killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero {α β : H → K} {x y : L}
    (hx : x ∈ rootSpace H α) (hy : y ∈ rootSpace H β) (hαβ : α + β ≠ 0) :
    killingForm K L x y = 0 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α β : ↥H → K} {x y : L} (hx : x ∈ LieAlgebra.rootSpace H α) (hy : y ∈ LieAlgebra.rootSpace H β) (hαβ : α + β ≠ 0)
Goal: ((killingForm K L) x) y = 0
-- LieAlgebra.mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_neg
lemma mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_neg
    {α : H → K} {x : L} (hx : x ∈ rootSpace H α)
    (hx' : ∀ y ∈ rootSpace H (-α), killingForm K L x y = 0) :
    x ∈ LinearMap.ker (killingForm K L) 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : ↥H → K} {x : L} (hx : x ∈ LieAlgebra.rootSpace H α) (hx' : ∀ y ∈ LieAlgebra.rootSpace H (-α), ((killingForm K L) x) y = 0)
Goal: x ∈ LinearMap.ker (killingForm K L)
-- LieAlgebra.IsKilling.eq_zero_of_isNilpotent_ad_of_mem_isCartanSubalgebra
lemma eq_zero_of_isNilpotent_ad_of_mem_isCartanSubalgebra {x : L} (hx : x ∈ H)
    (hx' : _root_.IsNilpotent (ad K L x)) : x = 0 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieAlgebra.IsKilling K L] {x : L} (hx : x ∈ H) (hx' : IsNilpotent ((LieAlgebra.ad K L) x))
Goal: x = 0
-- LieAlgebra.IsKilling.cartanEquivDual_apply_apply
@[simps! apply_apply]
noncomputable def cartanEquivDual :
    H ≃ₗ[K] Module.Dual K H 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieAlgebra.IsKilling K L]
Goal: ∀ (a m : ↥H),
  ((LieAlgebra.IsKilling.cartanEquivDual H) a) m =
    (LinearMap.trace K L) ((LieModule.toEnd K (↥H) L) a * (LieModule.toEnd K (↥H) L) m)
-- LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg_aux
lemma lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg_aux
    {α : Weight K H L} {e f : L} (heα : e ∈ rootSpace H α) (hfα : f ∈ rootSpace H (-α))
    (aux : ∀ (h : H), ⁅h, e⁆ = α h • e) :
    ⁅e, f⁆ = killingForm K L e f • (cartanEquivDual H).symm α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] {α : LieModule.Weight K (↥H) L} {e f : L} (heα : e ∈ LieAlgebra.rootSpace H ⇑α) (hfα : f ∈ LieAlgebra.rootSpace H (-⇑α)) (aux : ∀ (h : ↥H), ⁅h, e⁆ = α h • e)
Goal: ⁅e, f⁆ =
  ((killingForm K L) e) f • ↑((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α))
-- LieAlgebra.IsKilling.cartanEquivDual_symm_apply_mem_corootSpace
lemma cartanEquivDual_symm_apply_mem_corootSpace (α : Weight K H L) :
    (cartanEquivDual H).symm α ∈ corootSpace α 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] (α : LieModule.Weight K (↥H) L)
Goal: (LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α) ∈ LieAlgebra.corootSpace ⇑α
-- LieAlgebra.IsKilling.span_weight_eq_top
@[simp]
lemma span_weight_eq_top :
    span K (range (Weight.toLinear K H L)) = ⊤ 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L]
Goal: Submodule.span K (Set.range (LieModule.Weight.toLinear K (↥H) L)) = ⊤
-- LieAlgebra.IsKilling.span_weight_isNonZero_eq_top
@[simp]
lemma span_weight_isNonZero_eq_top :
    span K ({α : Weight K H L | α.IsNonZero}.image (Weight.toLinear K H L)) = ⊤ 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L]
Goal: Submodule.span K (LieModule.Weight.toLinear K (↥H) L '' {α | α.IsNonZero}) = ⊤