Datasets:

roseking
/

openAI-math-dataset

import OAI.Geometry.LatticeCovering.CircuitErrors

namespace OAI

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open MeasureTheory Filter Set
open scoped Topology
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open MeasureTheory Filter Set
open scoped Topology ENNReal
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open Module Submodule MeasureTheory
open scoped BigOperators

namespace SingleLatticeCovering.PrimeKernel
open LatticeGeometry Module Submodule MeasureTheory
open scoped BigOperators
variable {ι : Type*} [Fintype ι] [DecidableEq ι]

def scaleFactor (p : ℕ) : ℝ := (p : ℝ)^(-(Fintype.card ι : ℝ)⁻¹)
lemma scaleFactor_pos {ι : Type*} [Fintype ι] [DecidableEq ι] (p : ℕ) (hp : 0 < p) : 0 < scaleFactor (ι := ι) p :=
  Real.rpow_pos_of_pos (by exact_mod_cast hp) _

def normalizeEquiv (p : ℕ) (hp : 0 < p) : (ι → ℝ) ≃ₗ[ℝ] (ι → ℝ) :=
  LinearEquiv.smulOfNeZero ℝ (ι → ℝ) (scaleFactor (ι := ι) p) (scaleFactor_pos p hp).ne'
lemma normalizeEquiv_det (p : ℕ) (hp : 0 < p) (i : ι) :
    LinearMap.det (normalizeEquiv (ι := ι) p hp).toLinearMap=(p : ℝ)⁻¹ := by
  have hpR : (0 : ℝ)<p := by exact_mod_cast hp
  have hn : (Fintype.card ι : ℝ) ≠ 0 := by
    exact_mod_cast (Fintype.card_pos_iff.mpr ⟨i⟩).ne'
  change LinearMap.det (scaleFactor (ι := ι) p • (LinearMap.id : (ι → ℝ) →ₗ[ℝ] (ι → ℝ)))=_
  rw [LinearMap.det_smul,LinearMap.det_id,mul_one,Module.finrank_pi]
  rw [scaleFactor,←Real.rpow_natCast,←Real.rpow_mul hpR.le]
  have he : -(Fintype.card ι : ℝ)⁻¹*(Fintype.card ι : ℝ) = -1 := by field_simp
  rw [he,Real.rpow_neg_one]

abbrev normalized (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ) : Submodule ℤ (ι → ℝ) :=
  image (lattice p hp i a) (normalizeEquiv p hp)

lemma covolume_normalized (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ) :
    ZLattice.covolume (normalized p hp i a)=1 := by
  have hpR : (0 : ℝ)<p := by exact_mod_cast hp
  rw [image_covolume,normalizeEquiv_det p hp i,covolume_lattice,
    abs_of_pos (inv_pos.mpr hpR),inv_mul_cancel₀ hpR.ne']
def gridMap (p : ℕ) (hp : 0 < p) : (ι → ℤ) →+ (ι → ℝ) where
  toFun z := normalizeEquiv p hp (fun j => (z j : ℝ))
  map_zero' := by
    simp only [Pi.zero_apply,Int.cast_zero]
    exact map_zero (normalizeEquiv (ι := ι) p hp)
  map_add' z w := by
    change normalizeEquiv p hp (fun j => ((z+w) j : ℝ))=_
    rw [show (fun j => ((z+w) j : ℝ)) = (fun j => (z j : ℝ))+(fun j => (w j : ℝ)) by
      ext j; simp]
    exact map_add _ _ _
lemma gridMap_injective (p : ℕ) (hp : 0 < p) : Function.Injective (gridMap (ι := ι) p hp) := by
  intro z w h
  have he := (normalizeEquiv p hp).injective h
  ext j
  exact_mod_cast congrFun he j
def modForm (p : ℕ) (i : ι) (a : ι → ℤ) : (ι → ℤ) →+ ZMod p where
  toFun z := (z i : ZMod p)+∑ j ∈ Finset.univ.erase i, (a j : ZMod p)*(z j : ZMod p)
  map_zero' := by simp
  map_add' z w := by
    simp only [Pi.add_apply,Int.cast_add,mul_add,Finset.sum_add_distrib]
    ring
lemma grid_mem_normalized_iff (p : ℕ) (hp : 0 < p) (i : ι) (a z : ι → ℤ) :
    gridMap p hp z ∈ normalized p hp i a ↔ modForm p i a z=0 := by
  change (normalizeEquiv p hp) (fun j => (z j : ℝ)) ∈
    image (lattice p hp i a) (normalizeEquiv p hp) ↔ _
  change (normalizeEquiv p hp).symm ((normalizeEquiv p hp) (fun j => (z j : ℝ))) ∈ lattice p hp i a ↔ _
  rw [LinearEquiv.symm_apply_apply,integer_mem_iff_congruence]
  rfl
lemma exists_grid_of_mem (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ)
    {x : ι → ℝ} (hx : x ∈ normalized p hp i a) :
    ∃ z : ι → ℤ, gridMap p hp z=x ∧ modForm p i a z=0 := by
  obtain ⟨l,hl,he⟩ := (LatticeGeometry.mem_image _ _ _).mp hx
  obtain ⟨z,hz⟩ := exists_integer_of_mem p hp i a hl
  have hg : gridMap p hp z=x := by change normalizeEquiv p hp (fun j => (z j : ℝ))=x; rw [hz,he]
  exact ⟨z,hg,(grid_mem_normalized_iff p hp i a z).mp (hg.symm ▸ hx)⟩
def kernelEquiv (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ) :
    (modForm p i a).ker ≃+ normalized p hp i a :=
  AddEquiv.ofBijective
    ({ toFun := fun z => ⟨gridMap p hp z,(grid_mem_normalized_iff p hp i a z).mpr z.property⟩
       map_zero' := by apply Subtype.ext; exact map_zero _
       map_add' := fun z w => by apply Subtype.ext; exact map_add _ _ _ } :
      (modForm p i a).ker →+ normalized p hp i a)
    ⟨by intro z w h; apply Subtype.ext; exact gridMap_injective p hp (congrArg Subtype.val h),by
      intro x
      obtain ⟨z,hz,h0⟩ := exists_grid_of_mem p hp i a x.property
      exact ⟨⟨z,h0⟩,Subtype.ext hz⟩⟩
end SingleLatticeCovering.PrimeKernel
namespace SingleLatticeCovering.PrimeKernel
open Set MeasureTheory
open scoped Pointwise BigOperators ENNReal
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
def rectangularCell (p : ℕ) (i : ι) : Set (ι → ℝ) :=
  Set.univ.pi (fun j => Set.Ico (0:ℝ) (if j=i then (p:ℝ) else 1))
lemma mem_rectangularCell_iff {ι : Type*} [Fintype ι] [DecidableEq ι] (p : ℕ) (i : ι) (x : ι → ℝ) :
    x ∈ rectangularCell p i ↔ ∀ j, 0 ≤ x j ∧ x j < (if j=i then (p:ℝ) else 1) := by
  simp [rectangularCell,Set.mem_pi]
lemma rectangularCell_measurable (p : ℕ) (i : ι) : MeasurableSet (rectangularCell p i) :=
  MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Ico)
lemma integer_displacement_unique {p : ℕ} (hp : 0 < p) (i : ι)
    (a : ι → ℤ) {x y : ι → ℝ} (hx : x ∈ rectangularCell p i)
    (hy : y ∈ rectangularCell p i) (hxy : x-y ∈ lattice p hp i a) : x=y := by
  obtain ⟨z,hz,hdiv⟩ := (mem_lattice_iff p hp i a (x-y)).mp hxy
  have hzj : ∀ j, (z j:ℝ)=x j-y j := fun j => congrFun hz j
  have hxx := (mem_rectangularCell_iff p i x).mp hx
  have hyy := (mem_rectangularCell_iff p i y).mp hy
  have htail (j : ι) (hj : j ≠ i) : z j=0 := by
    have hxj := hxx j
    have hyj := hyy j
    rw [ite_eq_right hj] at hxj hyj
    have hlow : (-1:ℝ) < z j := by linarith [hzj j]
    have hupp : (z j:ℝ) < 1 := by linarith [hzj j]
    have hlowZ : (-1:ℤ) < z j := by exact_mod_cast hlow
    have huppZ : z j < 1 := by exact_mod_cast hupp
    omega
  have hsum : ∑ j ∈ Finset.univ.erase i, a j*z j = 0 := by
    apply Finset.sum_eq_zero
    intro j hj
    rw [htail j (Finset.mem_erase.mp hj).1,mul_zero]
  rw [hsum,add_zero] at hdiv
  obtain ⟨q,hq⟩ := hdiv
  have hxi := hxx i
  have hyi := hyy i
  simp at hxi hyi
  have hpR : (0:ℝ)<p := by exact_mod_cast hp
  have hzq : (z i:ℝ)=(p:ℝ)*(q:ℝ) := by exact_mod_cast hq
  have hqlo : (-1:ℝ) < q := by nlinarith [hzj i]
  have hqhi : (q:ℝ) < 1 := by nlinarith [hzj i]
  have hqloZ : (-1:ℤ) < q := by exact_mod_cast hqlo
  have hqhiZ : q < 1 := by exact_mod_cast hqhi
  have hq0 : q=0 := by omega
  have hzi : z i=0 := by rw [hq,hq0,mul_zero]
  ext j
  by_cases hj : j=i
  · subst j
    have hh := hzj i
    rw [hzi,Int.cast_zero] at hh
    linarith
  · have hh := hzj j
    rw [htail j hj,Int.cast_zero] at hh
    linarith
lemma rectangularCell_exists_shift (p : ℕ) (hp : 0 < p) (i : ι)
    (a : ι → ℤ) (x : ι → ℝ) :
    ∃ g : lattice p hp i a, (g:ι → ℝ)+x ∈ rectangularCell p i := by
  let f : ι → ℤ := fun j => ⌊x j⌋
  let T : ℤ := f i+∑ j ∈ Finset.univ.erase i, a j*f j
  let b : ℤ := T % (p:ℤ)
  let z : ι → ℤ := Function.update (-f) i (-f i+b)
  have hb0 : 0 ≤ b := Int.emod_nonneg _ (by exact_mod_cast hp.ne')
  have hbp : b < (p:ℤ) := Int.emod_lt_of_pos _ (by exact_mod_cast hp)
  have hzj (j : ι) (hj : j ≠ i) : z j = -f j := by simp [z,Function.update_of_ne hj]
  have hzi : z i = -f i+b := by simp [z]
  have hzlat : (fun j => (z j:ℝ)) ∈ lattice p hp i a := by
    apply (integer_mem_iff_dvd p hp i a z).mpr
    have hs : ∑ j ∈ Finset.univ.erase i, a j*z j = -(∑ j ∈ Finset.univ.erase i, a j*f j) := by
      rw [←Finset.sum_neg_distrib]
      apply Finset.sum_congr rfl
      intro j hj
      rw [hzj j (Finset.mem_erase.mp hj).1,mul_neg]
    rw [hzi,hs]
    have he : -f i+b-(∑ j ∈ Finset.univ.erase i, a j*f j) = -(T-b) := by dsimp [T]; ring
    change (p:ℤ) ∣ -f i+b-(∑ j ∈ Finset.univ.erase i, a j*f j)
    rw [he]
    apply dvd_neg.mpr
    exact Int.dvd_self_sub_emod
  refine ⟨⟨fun j => (z j:ℝ),hzlat⟩,?_⟩
  rw [mem_rectangularCell_iff]
  intro j
  have hflo := Int.floor_le (x j)
  have hfhi := Int.lt_floor_add_one (x j)
  change 0 ≤ (z j:ℝ)+x j ∧ (z j:ℝ)+x j < _
  change (f j:ℝ) ≤ x j at hflo
  change x j < (f j:ℝ)+1 at hfhi
  by_cases hj : j=i
  · subst j
    rw [hzi,Int.cast_add,Int.cast_neg,ite_eq_left rfl]
    have hbR : (0:ℝ)≤b := by exact_mod_cast hb0
    have hbpR : (b:ℝ)+1≤p := by exact_mod_cast (show b+1≤(p:ℤ) by omega)
    constructor <;> linarith
  · rw [hzj j hj,Int.cast_neg,ite_eq_right hj]
    constructor <;> linarith

lemma rectangularCell_fundamental (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ) :
    IsAddFundamentalDomain (lattice p hp i a) (rectangularCell p i) volume := by
  apply IsAddFundamentalDomain.mk' (rectangularCell_measurable p i).nullMeasurableSet
  intro x
  obtain ⟨g,hg⟩ := rectangularCell_exists_shift p hp i a x
  refine ⟨g,hg,?_⟩
  intro h hh
  apply Subtype.ext
  have he := integer_displacement_unique hp i a hh hg (by
    change ((h:ι → ℝ)+x)-((g:ι → ℝ)+x) ∈ lattice p hp i a
    simpa only [add_sub_add_right_eq_sub] using (lattice p hp i a).sub_mem h.property g.property)
  exact add_right_cancel he
end SingleLatticeCovering.PrimeKernel
namespace SingleLatticeCovering.PrimeKernel
open Set MeasureTheory
open scoped Pointwise BigOperators ENNReal
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
abbrev normalizedRectangularCell (p : ℕ) (hp : 0 < p) (i : ι) : Set (ι → ℝ) :=
  normalizeEquiv p hp '' rectangularCell p i
lemma normalizedRectangularCell_fundamental (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ) :
    IsAddFundamentalDomain (normalized p hp i a) (normalizedRectangularCell p hp i) volume := by
  let e := normalizeEquiv (ι:=ι) p hp
  let eg : normalized p hp i a ≃ lattice p hp i a :=
    { toFun := fun x => ⟨e.symm x,x.property⟩
      invFun := fun x => ⟨e x,by change e.symm (e x) ∈ lattice p hp i a; simp⟩
      left_inv := fun x => by apply Subtype.ext; exact e.apply_symm_apply x
      right_inv := fun x => by apply Subtype.ext; exact e.symm_apply_apply x }
  have he : LinearMap.det e.symm.toLinearMap ≠ 0 := (LinearEquiv.isUnit_det' e.symm).ne_zero
  have hf := Measure.LinearMap.quasiMeasurePreserving volume e.symm.toLinearMap he
  apply (rectangularCell_fundamental p hp i a).image_of_equiv e.toEquiv hf eg
  intro g x
  change e ((e.symm (g:ι → ℝ))+x)=(g:ι → ℝ)+e x
  rw [map_add,e.apply_symm_apply]

lemma rectangularCell_bounded (p : ℕ) (i : ι) : Bornology.IsBounded (rectangularCell p i) := by
  apply Bornology.IsBounded.subset (isCompact_Icc : IsCompact (Set.Icc (0:ι → ℝ) (fun j => if j=i then (p:ℝ) else 1))).isBounded
  intro x hx
  rw [mem_rectangularCell_iff] at hx
  exact ⟨fun j => (hx j).1,fun j => (hx j).2.le⟩

lemma normalizedRectangularCell_measurable (p : ℕ) (hp : 0 < p) (i : ι) :
    MeasurableSet (normalizedRectangularCell p hp i) :=
  (normalizeEquiv p hp).toContinuousLinearEquiv.toHomeomorph.measurableEmbedding.measurableSet_image.mpr
    (rectangularCell_measurable p i)

lemma volume_rectangularCell (p : ℕ) (i : ι) : volume (rectangularCell p i)=(p:ℝ≥0∞) := by
  classical
  rw [rectangularCell,volume_pi,Measure.pi_pi]
  simp only [Real.volume_Ico,sub_zero]
  rw [Finset.prod_eq_single i]
  · simp
  · intro j hj hji
    simp [hji]
  · simp

lemma volume_normalizedRectangularCell (p : ℕ) (hp : 0 < p) (i : ι) :
    volume (normalizedRectangularCell p hp i)=1 := by
  change volume ((normalizeEquiv (ι:=ι) p hp).toContinuousLinearEquiv '' rectangularCell p i)=1
  rw [Measure.addHaar_image_continuousLinearEquiv]
  change ENNReal.ofReal |LinearMap.det (normalizeEquiv (ι:=ι) p hp).toLinearMap| * volume (rectangularCell p i)=1
  rw [normalizeEquiv_det p hp i,
    abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _)),volume_rectangularCell]
  rw [ENNReal.ofReal_inv_of_pos (by exact_mod_cast hp),ENNReal.ofReal_natCast]
  exact ENNReal.inv_mul_cancel (by exact_mod_cast hp.ne') (by simp)
end SingleLatticeCovering.PrimeKernel
namespace SingleLatticeCovering.PrimeKernel
open Set MeasureTheory
open scoped Pointwise BigOperators ENNReal
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
def gridUnitCell : Set (ι → ℝ) := Set.univ.pi (fun _ => Set.Ico (0:ℝ) 1)
def offsetVector (i : ι) (b : ℤ) : ι → ℤ := Pi.single i b
def offsetCell (p : ℕ) (hp : 0 < p) (i : ι) (b : Fin p) : Set (ι → ℝ) :=
  normalizeEquiv p hp '' ((fun j => ((offsetVector i b.val j:ℤ):ℝ)) +ᵥ gridUnitCell)
lemma volume_gridUnitCell {ι : Type*} [Fintype ι] [DecidableEq ι] : volume (gridUnitCell (ι:=ι))=1 := by
  rw [gridUnitCell,volume_pi,Measure.pi_pi]
  simp
lemma volume_offsetCell (p : ℕ) (hp : 0 < p) (i : ι) (b : Fin p) :
    volume (offsetCell p hp i b)=(p:ℝ≥0∞)⁻¹ := by
  change volume ((normalizeEquiv (ι:=ι) p hp).toContinuousLinearEquiv '' _)=_
  rw [Measure.addHaar_image_continuousLinearEquiv]
  change ENNReal.ofReal |LinearMap.det (normalizeEquiv (ι:=ι) p hp).toLinearMap| * volume _ = _
  rw [normalizeEquiv_det p hp i,abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _)),
    measure_vadd,volume_gridUnitCell,mul_one,ENNReal.ofReal_inv_of_pos (by exact_mod_cast hp),
    ENNReal.ofReal_natCast]
lemma rectangularCell_unit_decomposition (p : ℕ) (hp : 0 < p) (i : ι)
    {x : ι → ℝ} (hx : x ∈ rectangularCell p i) :
    ∃ b : Fin p, ∃ u ∈ gridUnitCell, x=(fun j => ((offsetVector i b.val j:ℤ):ℝ))+u := by
  have _ := hp
  have hxj := (mem_rectangularCell_iff p i x).mp hx
  have hxi : 0 ≤ x i ∧ x i < (p:ℝ) := by simpa using hxj i
  let q : ℤ := ⌊x i⌋
  have hq0 : 0 ≤ q := Int.floor_nonneg.mpr hxi.1
  have hqp : q < (p:ℤ) := Int.floor_lt.mpr hxi.2
  let b : Fin p := ⟨q.toNat,by omega⟩
  have hbq : (b.val:ℤ)=q := Int.toNat_of_nonneg hq0
  let o : ι → ℝ := fun j => ((offsetVector i b.val j:ℤ):ℝ)
  refine ⟨b,x-o,?_,by dsimp [o]; abel⟩
  rw [gridUnitCell,Set.mem_pi]
  intro j hj
  change 0 ≤ x j-o j ∧ x j-o j < 1
  by_cases hji : j=i
  · subst j
    have hoi : o i=(q:ℝ) := by simp [o,offsetVector,hbq]
    rw [hoi]
    have hl := Int.floor_le (x i)
    have hu := Int.lt_floor_add_one (x i)
    change (q:ℝ) ≤ x i at hl
    change x i < (q:ℝ)+1 at hu
    constructor <;> linarith
  · have hoj : o j=0 := by simp [o,offsetVector,Pi.single_eq_of_ne hji]
    rw [hoj,sub_zero]
    simpa [hji] using hxj j
lemma normalizedRectangularCell_subset_offsetCells (p : ℕ) (hp : 0 < p) (i : ι) :
    normalizedRectangularCell p hp i ⊆ ⋃ b : Fin p, offsetCell p hp i b := by
  rintro _ ⟨x,hx,rfl⟩
  obtain ⟨b,u,hu,he⟩ := rectangularCell_unit_decomposition p hp i hx
  apply Set.mem_iUnion.mpr
  refine ⟨b,?_⟩
  apply Set.mem_image_of_mem
  exact ⟨u,hu,he.symm⟩
lemma modForm_offsetVector (p : ℕ) (i : ι) (a : ι → ℤ) (b : ℤ) :
    modForm p i a (offsetVector i b)=(b:ZMod p) := by
  change ((offsetVector i b i:ℤ):ZMod p)+∑ j ∈ Finset.univ.erase i,
    (a j:ZMod p)*((offsetVector i b j:ℤ):ZMod p)=_
  have hs : ∑ j ∈ Finset.univ.erase i, (a j:ZMod p)*((offsetVector i b j:ℤ):ZMod p)=0 := by
    apply Finset.sum_eq_zero
    intro j hj
    simp [offsetVector,Pi.single_eq_of_ne (Finset.mem_erase.mp hj).1]
  rw [hs,add_zero]
  simp [offsetVector]
lemma offsetCell_hit (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ)
    (S : Finset (ι → ℤ)) {J : Set (ι → ℝ)}
    (hthick : ∀ z ∈ S, ∀ u ∈ gridUnitCell, gridMap p hp z + normalizeEquiv p hp u ∈ J)
    (b : Fin p) (hb : ∃ z ∈ S, modForm p i a z=(b.val:ZMod p))
    {x : ι → ℝ} (hx : x ∈ offsetCell p hp i b) :
    ∃ l : normalized p hp i a, x+l ∈ J := by
  obtain ⟨z,hz,hzb⟩ := hb
  obtain ⟨y,⟨u,hu,rfl⟩,rfl⟩ := hx
  let w := z-offsetVector i b.val
  have hw : gridMap p hp w ∈ normalized p hp i a := by
    rw [grid_mem_normalized_iff]
    dsimp [w]
    rw [map_sub,modForm_offsetVector,hzb]
    simp
  refine ⟨⟨gridMap p hp w,hw⟩,?_⟩
  have hh := hthick z hz u hu
  convert hh using 1
  change normalizeEquiv p hp ((fun j => ((offsetVector i b.val j:ℤ):ℝ))+u)+
    normalizeEquiv p hp (fun j => (w j:ℝ)) =
      normalizeEquiv p hp (fun j => (z j:ℝ))+normalizeEquiv p hp u
  rw [←map_add,←map_add]
  apply congrArg (normalizeEquiv p hp)
  ext j
  simp only [Pi.add_apply,w,Pi.sub_apply,Int.cast_sub]
  ring
end SingleLatticeCovering.PrimeKernel
namespace SingleLatticeCovering.PrimeKernel
open Set MeasureTheory
open scoped Pointwise BigOperators ENNReal
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
noncomputable def badOffsets (p : ℕ) (i : ι) (a : ι → ℤ) (S : Finset (ι → ℤ)) : Finset (Fin p) := by
  classical
  exact Finset.univ.filter (fun b => ∀ z ∈ S, modForm p i a z ≠ (b.val:ZMod p))
lemma cell_hole_le_badOffsets (p : ℕ) (hp : 0 < p) (i : ι) (a : ι → ℤ)
    (S : Finset (ι → ℤ)) {J : Set (ι → ℝ)}
    (hthick : ∀ z ∈ S, ∀ u ∈ gridUnitCell, gridMap p hp z + normalizeEquiv p hp u ∈ J) :
    volume (normalizedRectangularCell p hp i ∩ {x | ∀ l : normalized p hp i a, x+l ∉ J}) ≤
      ((badOffsets p i a S).card:ℝ≥0∞)/(p:ℝ≥0∞) := by
  classical
  have hsub : normalizedRectangularCell p hp i ∩ {x | ∀ l : normalized p hp i a, x+l ∉ J} ⊆
      ⋃ b ∈ badOffsets p i a S, offsetCell p hp i b := by
    rintro x ⟨hx,hmiss⟩
    obtain ⟨b,hb⟩ := Set.mem_iUnion.mp (normalizedRectangularCell_subset_offsetCells p hp i hx)
    have hbad : b ∈ badOffsets p i a S := by
      simp only [badOffsets,Finset.mem_filter,Finset.mem_univ,true_and]
      intro z hz hzb
      obtain ⟨l,hl⟩ := offsetCell_hit p hp i a S hthick b ⟨z,hz,hzb⟩ hb
      exact hmiss l hl
    exact Set.mem_iUnion.mpr ⟨b,Set.mem_iUnion.mpr ⟨hbad,hb⟩⟩
  calc
    _ ≤ volume (⋃ b ∈ badOffsets p i a S, offsetCell p hp i b) := measure_mono hsub
    _ ≤ ∑ b ∈ badOffsets p i a S, volume (offsetCell p hp i b) := measure_biUnion_finset_le _ _
    _ = _ := by simp [volume_offsetCell,div_eq_mul_inv]
end SingleLatticeCovering.PrimeKernel
namespace SingleLatticeCovering.RogersPreparation
open Horizontal Completion Set MeasureTheory
open scoped Pointwise ENNReal
lemma pointCount_zero_iff_no_hit {d : ℕ} (L : FullLattice d)
    (J : Set (Fin d → ℝ)) (x : Fin d → ℝ) :
    pointCount L J x=0 ↔ ∀ l : L.module, x+l ∉ J := by
  classical
  simp only [pointCount,periodize,ENNReal.tsum_eq_zero]
  apply forall_congr'
  intro l
  by_cases h : x+l ∈ J <;> simp [h]



lemma hole_eq_unit_fundamental_hole {d : ℕ} (L : FullLattice d)
    {J F : Set (Fin d → ℝ)} (hJ : IsCompact J)
    (hF : IsAddFundamentalDomain L.module F volume) (hvol : volume F=1) :
    L.hole J=volume (F ∩ {x | ∀ l : L.module, x+l ∉ J}) := by
  let H : Set (Fin d → ℝ) := {x | pointCount L J x=0}
  have hH : MeasurableSet H := (measurable_periodize L (measurable_one.indicator hJ.measurableSet)) (.singleton 0)
  have hperiodic (l : L.module) : (fun x => (l:Fin d → ℝ)+x) ⁻¹' H=H := by
    ext x
    change pointCount L J ((l:Fin d → ℝ)+x)=0 ↔ pointCount L J x=0
    rw [pointCount_eq_zero,pointCount_eq_zero,map_add,
      (latticeProjection_eq_zero_iff L.module _).mpr l.property,zero_add]
  have hv : volume (cell L)=1 := (cell_fundamental L).measure_eq hF |>.trans hvol
  rw [hole_eq_zero_count_probability L hJ,translateLaw,hv,inv_one,one_smul,
    Measure.restrict_apply hH]
  change volume (H ∩ cell L)=_
  rw [(cell_fundamental L).measure_set_eq hF hH hperiodic]
  congr 1
  ext x
  simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,H,pointCount_zero_iff_no_hit,and_comm]
end SingleLatticeCovering.RogersPreparation
namespace SingleLatticeCovering.PrimeKernel
open RogersPreparation Horizontal Set MeasureTheory
open scoped Pointwise ENNReal
variable {d : ℕ}
def normalizedFull (p : ℕ) (hp : 0 < p) (i : Fin d) (a : Fin d → ℤ) : FullLattice d :=
  ⟨normalized p hp i a,inferInstance,inferInstance⟩
lemma normalizedFull_hole_le_badOffsets (p : ℕ) (hp : 0 < p) (i : Fin d) (a : Fin d → ℤ)
    (S : Finset (Fin d → ℤ)) {J : Set (Fin d → ℝ)} (hJ : IsCompact J)
    (hthick : ∀ z ∈ S, ∀ u ∈ gridUnitCell, gridMap p hp z + normalizeEquiv p hp u ∈ J) :
    (normalizedFull p hp i a).hole J ≤ ((badOffsets p i a S).card:ℝ≥0∞)/(p:ℝ≥0∞) := by
  rw [hole_eq_unit_fundamental_hole (normalizedFull p hp i a) hJ
    (normalizedRectangularCell_fundamental p hp i a) (volume_normalizedRectangularCell p hp i)]
  exact cell_hole_le_badOffsets p hp i a S hthick
end SingleLatticeCovering.PrimeKernel
noncomputable section
namespace SingleLatticeCovering.AffineVoid
open scoped BigOperators
variable {K : Type*} [Field K] [Fintype K] [DecidableEq K]
variable {n : ℕ} {σ : Type*} [Fintype σ]
def normalDual (c : Kˣ) (b : K) (a : Fin n → K) : Fin (n+2) → K :=
  Fin.cons (-(c:K)*b) (Fin.cons (c:K) (fun j => (c:K)*a j))
def fullMiss (v : σ → Fin (n+1) → K) (w : Fin (n+2) → K) : ℝ := by
  classical
  exact if ∀ i, (∑ j, w j * (Fin.cons (1:K) (v i) : Fin (n+2) → K) j) ≠ 0 then 1 else 0
def normalizedMiss (v : σ → Fin (n+1) → K) (a : Fin n → K) (b : K) : ℝ := by
  classical
  exact if ∀ i, v i 0 + ∑ j, a j*v i j.succ ≠ b then 1 else 0
lemma fullMiss_nonneg {K : Type*} [Field K] [Fintype K] [DecidableEq K] {n : ℕ} {σ : Type*} [Fintype σ] (v : σ → Fin (n+1) → K) (w : Fin (n+2) → K) :
    0 ≤ fullMiss v w := by
  classical
  unfold fullMiss
  split_ifs <;> norm_num
lemma normalizedMiss_nonneg {K : Type*} [Field K] [Fintype K] [DecidableEq K] {n : ℕ} {σ : Type*} [Fintype σ] (v : σ → Fin (n+1) → K) (a : Fin n → K) (b : K) :
    0 ≤ normalizedMiss v a b := by
  classical
  unfold normalizedMiss
  split_ifs <;> norm_num
lemma normalDual_injective {K : Type*} [Field K] [Fintype K] [DecidableEq K] {n : ℕ} : Function.Injective
    (fun t : Kˣ × K × (Fin n → K) => normalDual t.1 t.2.1 t.2.2) := by
  rintro ⟨c,b,a⟩ ⟨c',b',a'⟩ he
  have hc : (c:K)=(c':K) := by
    have h := congrFun he ((0:Fin (n+1)).succ)
    simpa [normalDual] using h
  have hcc : c=c' := Units.ext hc
  subst c'
  have hb : b=b' := by
    have h := congrFun he 0
    simp only [normalDual,Fin.cons_zero] at h
    exact mul_left_cancel₀ (neg_ne_zero.mpr c.ne_zero) h
  have ha : a=a' := by
    funext j
    have h := congrFun he j.succ.succ
    simp only [normalDual,Fin.cons_succ] at h
    exact mul_left_cancel₀ c.ne_zero h
  subst b'; subst a'; rfl
lemma normalDual_dot {K : Type*} [Field K] [Fintype K] [DecidableEq K] {n : ℕ} (c : Kˣ) (b : K) (a : Fin n → K) (z : Fin (n+1) → K) :
    (∑ j, normalDual c b a j * (Fin.cons (1:K) z : Fin (n+2) → K) j) =
      (c:K)*(z 0+∑ j, a j*z j.succ-b) := by
  simp only [normalDual,Fin.sum_univ_succ,Fin.cons_zero,Fin.cons_succ,mul_one]
  rw [mul_sub,mul_add,Finset.mul_sum]
  ring_nf
lemma fullMiss_normalDual (v : σ → Fin (n+1) → K) (c : Kˣ) (b : K) (a : Fin n → K) :
    fullMiss v (normalDual c b a)=normalizedMiss v a b := by
  classical
  unfold fullMiss normalizedMiss
  have he : (∀ i, (∑ j, normalDual c b a j * (Fin.cons (1:K) (v i) : Fin (n+2) → K) j) ≠ 0) ↔
      (∀ i, v i 0+∑ j, a j*v i j.succ ≠ b) := by
    apply forall_congr'
    intro i
    rw [normalDual_dot]
    exact ⟨fun h => sub_ne_zero.mp (mul_ne_zero_iff.mp h).2,
      fun h => mul_ne_zero c.ne_zero (sub_ne_zero.mpr h)⟩
  simp only [he]



lemma raw_conditioning_bound (v : σ → Fin (n+1) → K) :
    (Fintype.card Kˣ:ℝ)*(∑ a : Fin n → K, ∑ b : K, normalizedMiss v a b) ≤
      ∑ w : Fin (n+2) → K, fullMiss v w := by
  classical
  let f := fun t : Kˣ × K × (Fin n → K) => normalDual t.1 t.2.1 t.2.2
  have hs : (∑ t, fullMiss v (f t)) ≤ ∑ w, fullMiss v w := by
    rw [←Finset.sum_image (fun x hx y hy he => normalDual_injective he)]
    exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
      (fun w hw hwt => fullMiss_nonneg v w)
  have he : (∑ t, fullMiss v (f t)) =
      (Fintype.card Kˣ:ℝ)*(∑ a : Fin n → K, ∑ b : K, normalizedMiss v a b) := by
    simp only [f,Fintype.sum_prod_type,fullMiss_normalDual]
    rw [Finset.sum_const,nsmul_eq_mul]
    congr 1
    exact Finset.sum_comm
  rwa [he] at hs

def meanFull (v : σ → Fin (n+1) → K) : ℝ :=
  (∑ w : Fin (n+2) → K, fullMiss v w)/(Fintype.card K:ℝ)^(n+2)
def meanNormalized (v : σ → Fin (n+1) → K) : ℝ :=
  (∑ a : Fin n → K, ∑ b : K, normalizedMiss v a b)/(Fintype.card K:ℝ)^(n+1)

lemma conditioning_bound (v : σ → Fin (n+1) → K) :
    meanNormalized v ≤ ((Fintype.card K:ℝ)/((Fintype.card K:ℝ)-1))*meanFull v := by
  have hcard : 1 < Fintype.card K := Fintype.one_lt_card
  have hc : (1:ℝ) < Fintype.card K := by exact_mod_cast hcard
  have hc0 : (0:ℝ) < Fintype.card K := by linarith
  have hcm : (0:ℝ) < (Fintype.card K:ℝ)-1 := by linarith
  have hh := raw_conditioning_bound v
  rw [Fintype.card_units,Nat.cast_sub (by omega),Nat.cast_one] at hh
  unfold meanNormalized meanFull
  apply (div_le_iff₀ (pow_pos hc0 _)).mpr
  calc
    _ ≤ (∑ w : Fin (n+2) → K, fullMiss v w)/((Fintype.card K:ℝ)-1) :=
      (le_div_iff₀ hcm).mpr (by simpa only [mul_comm] using hh)
    _ = _ := by
      rw [show (Fintype.card K:ℝ)^(n+2)=(Fintype.card K:ℝ)^(n+1)*(Fintype.card K:ℝ) by rw [show n+2=(n+1)+1 by omega,pow_succ]]
      field_simp [hc0.ne',hcm.ne']


end SingleLatticeCovering.AffineVoid


noncomputable section
namespace SingleLatticeCovering.ConvexCore
open Set MeasureTheory Filter Topology Metric
open scoped Pointwise ENNReal
variable {d : ℕ}

lemma interior_nonempty_of_volume_pos {J : Set (Fin d → ℝ)}
    (hconv : Convex ℝ J) (hV : 0 < volume.real J) : (interior J).Nonempty := by
  by_contra hh
  have hi : interior J = ∅ := Set.not_nonempty_iff_eq_empty.mp hh
  have hs : J ⊆ frontier J := by
    rw [frontier,hi,sdiff_empty]
    exact subset_closure
  have hz : volume J = 0 := measure_mono_null hs (hconv.addHaar_frontier volume)
  simp only [measureReal_def,hz,ENNReal.toReal_zero] at hV
  exact lt_irrefl _ hV

lemma core_subset_interior {J : Set (Fin d → ℝ)} (hconv : Convex ℝ J)
    (h0 : 0 ∈ interior J) {c : ℝ} (hc0 : 0 ≤ c) (hc1 : c < 1) :
    c • J ⊆ interior J := by
  rintro x ⟨y,hy,rfl⟩
  simpa only [smul_zero,zero_add] using
    hconv.combo_interior_self_mem_interior h0 hy (sub_pos.mpr hc1) hc0 (sub_add_cancel 1 c)

lemma core_compact {J : Set (Fin d → ℝ)} (hJ : IsCompact J) (c : ℝ) :
    IsCompact (c • J) := by
  exact hJ.image (by fun_prop)

lemma core_volume {J : Set (Fin d → ℝ)} {c : ℝ} (hc : 0 ≤ c) :
    volume.real (c • J)=c^d*volume.real J := by
  rw [measureReal_def,Measure.addHaar_smul_of_nonneg volume hc,ENNReal.toReal_mul,
    ENNReal.toReal_ofReal (pow_nonneg hc _)]
  simp only [Module.finrank_pi,Fintype.card_fin]
  rfl

lemma compact_uniform_add {J C : Set (Fin d → ℝ)} (hC : IsCompact C)
    (hsub : C ⊆ interior J) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ x ∈ C, ∀ u : Fin d → ℝ, ‖u‖ ≤ δ → x+u ∈ J := by
  obtain ⟨δ,hδ,hs⟩ := hC.exists_cthickening_subset_open isOpen_interior hsub
  refine ⟨δ,hδ,fun x hx u hu => interior_subset (hs ?_)⟩
  apply Metric.mem_cthickening_of_dist_le (x+u) x δ C hx
  simpa only [dist_eq_norm,add_sub_cancel_left] using hu

lemma exists_shrink_exp {V ε : ℝ} (hε : 0 < ε) :
    ∃ c : ℝ, 0 < c ∧ c < 1 ∧ Real.exp (-(c^d*V)) < Real.exp (-V)+ε := by
  have hc : ContinuousAt (fun c : ℝ => Real.exp (-(c^d*V))) 1 := by fun_prop
  obtain ⟨δ,hδ,hclose⟩ := Metric.continuousAt_iff.mp hc ε hε
  let t := min (δ/2) (1/2:ℝ)
  have ht : 0 < t := lt_min (half_pos hδ) (by norm_num)
  have htδ : t < δ := (min_le_left _ _).trans_lt (half_lt_self hδ)
  have ht1 : t < 1 := (min_le_right _ _).trans_lt (by norm_num)
  refine ⟨1-t,by linarith,by linarith,?_⟩
  have hh := hclose (show dist (1-t) 1 < δ by
    simpa only [Real.dist_eq,sub_sub_cancel_left,abs_neg,abs_of_pos ht] using htδ)
  simp only [one_pow,one_mul,Real.dist_eq] at hh
  linarith [(abs_lt.mp hh).2]

end SingleLatticeCovering.ConvexCore


end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI