# Documentation

Mathlib.Algebra.Module.Zlattice

# ℤ-lattices #

Let E be a finite dimensional vector space over a NormedLinearOrderedField K with a solid norm that is also a FloorRing, e.g. ℝ. A (full) ℤ-lattice L of E is a discrete subgroup of E such that L spans E over K.

A ℤ-lattice L can be defined in two ways:

• For b a basis of E, then L = Submodule.span ℤ (Set.range b) is a ℤ-lattice of E
• As an AddSubgroup E with the additional properties:
• DiscreteTopology L, that is L is discrete
• Submodule.span ℝ (L : Set E) = ⊤, that is L spans E over K.

Results about the first point of view are in the Zspan namespace and results about the second point of view are in the Zlattice namespace.

## Main results #

• Zspan.isAddFundamentalDomain: for a ℤ-lattice Submodule.span ℤ (Set.range b), proves that the set defined by Zspan.fundamentalDomain is a fundamental domain.
• Zlattice.module_free: an AddSubgroup of E that is discrete and spans E over K is a free ℤ-module
• Zlattice.rank: an AddSubgroup of E that is discrete and spans E over K is a free ℤ-module of ℤ-rank equal to the K-rank of E
def Zspan.fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) :
Set E

The fundamental domain of the ℤ-lattice spanned by b. See Zspan.isAddFundamentalDomain for the proof that it is a fundamental domain.

Instances For
@[simp]
theorem Zspan.mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) {m : E} :
∀ (i : ι), ↑(b.repr m) i Set.Ico 0 1
theorem Zspan.map_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) {F : Type u_4} [] (f : E ≃ₗ[K] F) :
@[simp]
theorem Zspan.fundamentalDomain_reindex {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) {ι' : Type u_4} (e : ι ι') :
theorem Zspan.fundamentalDomain_pi_basisFun {ι : Type u_2} [] :
= Set.pi Set.univ fun x => Set.Ico 0 1
def Zspan.floor {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) :
{ x // x }

The map that sends a vector of E to the element of the ℤ-lattice spanned by b obtained by rounding down its coordinates on the basis b.

Instances For
def Zspan.ceil {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) :
{ x // x }

The map that sends a vector of E to the element of the ℤ-lattice spanned by b obtained by rounding up its coordinates on the basis b.

Instances For
@[simp]
theorem Zspan.repr_floor_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) (i : ι) :
↑(b.repr ↑()) i = ↑(b.repr m) i
@[simp]
theorem Zspan.repr_ceil_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) (i : ι) :
↑(b.repr ↑()) i = ↑(b.repr m) i
@[simp]
theorem Zspan.floor_eq_self_of_mem {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) (h : m ) :
↑() = m
@[simp]
theorem Zspan.ceil_eq_self_of_mem {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) (h : m ) :
↑() = m
def Zspan.fract {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) :
E

The map that sends a vector E to the fundamentalDomain of the lattice, see Zspan.fract_mem_fundamentalDomain, and fractRestrict for the map with the codomain restricted to fundamentalDomain.

Instances For
theorem Zspan.fract_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) :
= m - ↑()
@[simp]
theorem Zspan.repr_fract_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) (i : ι) :
↑(b.repr ()) i = Int.fract (↑(b.repr m) i)
@[simp]
theorem Zspan.fract_fract {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) :
@[simp]
theorem Zspan.fract_zspan_add {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) {v : E} (h : v ) :
Zspan.fract b (v + m) =
@[simp]
theorem Zspan.fract_add_zspan {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) {v : E} (h : v ) :
Zspan.fract b (m + v) =
theorem Zspan.fract_eq_self {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] {b : Basis ι K E} [] [] {x : E} :
= x
theorem Zspan.fract_mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (x : E) :
def Zspan.fractRestrict {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (x : E) :

The map fract with codomain restricted to fundamentalDomain.

Instances For
theorem Zspan.fractRestrict_surjective {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] :
@[simp]
theorem Zspan.fractRestrict_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (x : E) :
↑() =
theorem Zspan.fract_eq_fract {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (m : E) (n : E) :
= -m + n
theorem Zspan.norm_fract_le {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] [] (m : E) :
Finset.sum Finset.univ fun i => b i
@[simp]
theorem Zspan.coe_floor_self {ι : Type u_2} {K : Type u_3} [] [] [] (k : K) :
↑(Zspan.floor () k) = k
@[simp]
theorem Zspan.coe_fract_self {ι : Type u_2} {K : Type u_3} [] [] [] (k : K) :
theorem Zspan.fundamentalDomain_isBounded {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] [] :
theorem Zspan.vadd_mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (y : { x // x }) (x : E) :
y = -
theorem Zspan.exist_unique_vadd_mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (x : E) :
∃! v,
def Zspan.quotientEquiv {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] :
E

The map Zspan.fractRestrict defines an equiv map between E ⧸ span ℤ (Set.range b) and Zspan.fundamentalDomain b.

Instances For
@[simp]
theorem Zspan.quotientEquiv_apply_mk {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (x : E) :
@[simp]
theorem Zspan.quotientEquiv.symm_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [] (b : Basis ι K E) [] [] (x : ) :
().symm x =
theorem Zspan.fundamentalDomain_measurableSet {E : Type u_1} {ι : Type u_2} [] (b : Basis ι E) [] [] :
theorem Zspan.isAddFundamentalDomain {E : Type u_1} {ι : Type u_2} [] (b : Basis ι E) [] [] (μ : ) :

For a ℤ-lattice Submodule.span ℤ (Set.range b), proves that the set defined by Zspan.fundamentalDomain is a fundamental domain.

theorem Zspan.measure_fundamentalDomain {E : Type u_1} {ι : Type u_2} [] (b : Basis ι E) [] [] [] (μ : ) [] (b₀ : Basis ι E) :
μ () = ENNReal.ofReal |Matrix.det (Basis.toMatrix b₀ b)| * μ ()
@[simp]
theorem Zspan.volume_fundamentalDomain {ι : Type u_2} [] [] (b : Basis ι (ι)) :
MeasureTheory.volume () = ENNReal.ofReal |Matrix.det (Matrix.of b)|
theorem Zlattice.FG (K : Type u_1) [] [] {E : Type u_2} [] [] [] {L : } [DiscreteTopology { x // x L }] (hs : = ) :
theorem Zlattice.module_finite (K : Type u_1) [] [] {E : Type u_2} [] [] [] {L : } [DiscreteTopology { x // x L }] (hs : = ) :
Module.Finite { x // x L }
theorem Zlattice.module_free (K : Type u_1) [] [] {E : Type u_2} [] [] [] {L : } [DiscreteTopology { x // x L }] (hs : = ) :
Module.Free { x // x L }
theorem Zlattice.rank (K : Type u_1) [] [] {E : Type u_2} [] [] [] {L : } [DiscreteTopology { x // x L }] (hs : = ) :