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import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
namespace OAI
universe uι
open Set MeasureTheory
open scoped ENNReal Pointwise
namespace Paper092
variable {ι : Type uι} [Fintype ι] [DecidableEq ι]
def unitCube (ι : Type uι) : Set (ι → ℝ) := Icc 0 1
def extrusionCell (i : ι) : Set (ι → ℝ) :=
{x | (1 ≤ x i ∧ x i ≤ 2) ∧ ∀ j, j ≠ i → x i - 1 ≤ x j ∧ x j ≤ x i}
def shearMatrix (i : ι) : Matrix ι ι ℝ :=
(1 : Matrix ι ι ℝ).updateCol i (fun _ => 1)
theorem det_shearMatrix (i : ι) : (shearMatrix i).det = 1 := by
simpa [shearMatrix, Matrix.one_apply] using
Matrix.det_updateCol_sum (1 : Matrix ι ι ℝ) i (fun _ => (1 : ℝ))
def cellMap (i : ι) (x : ι → ℝ) : ι → ℝ :=
fun j => if j = i then 1 + x i else x i + x j
omit [Fintype ι] in
theorem cellMap_mem_iff (i : ι) (x : ι → ℝ) :
cellMap i x ∈ extrusionCell i ↔ x ∈ unitCube ι := by
simp only [extrusionCell, cellMap, unitCube, mem_ofPred_eq, mem_Icc, Pi.le_def,
Pi.zero_apply, Pi.one_apply, ite_true]
constructor
· rintro ⟨hi, h⟩
constructor <;> intro j
· by_cases hj : j = i
· subst j; linarith
· have := h j hj
simp only [ite_eq_right hj] at this
linarith
· by_cases hj : j = i
· subst j; linarith
· have := h j hj
simp only [ite_eq_right hj] at this
linarith
· rintro ⟨h0, h1⟩
refine ⟨⟨by linarith [h0 i], by linarith [h1 i]⟩, ?_⟩
intro j hj
simp only [ite_eq_right hj]
constructor <;> linarith [h0 j, h1 j]
theorem cellMap_eq_matrix (i : ι) (x : ι → ℝ) :
cellMap i x = Pi.single i 1 + Matrix.toLin' (shearMatrix i) x := by
funext j
simp only [cellMap, Pi.add_apply, Matrix.toLin'_apply, Matrix.mulVec, dotProduct,
shearMatrix, Matrix.updateCol_apply, Matrix.one_apply]
by_cases hj : j = i
· subst j
have hsum : ∀ k : ι,
(if k = i then 1 else if i = k then 1 else 0) * x k =
if k = i then x k else 0 := by
intro k
by_cases hk : k = i
· subst k; simp
· simp [hk, Ne.symm hk]
simp_rw [hsum]
simp
· have hsum : ∀ k : ι,
(if k = i then 1 else if j = k then 1 else 0) * x k =
(if k = i then x i else 0) + (if k = j then x j else 0) := by
intro k
by_cases hki : k = i
· subst k; simp [Ne.symm hj]
· by_cases hkj : k = j
· subst k; simp [hj]
· simp [hki, hkj, Ne.symm hkj]
simp_rw [hsum]
simp [hj, Finset.sum_add_distrib]
theorem cellMap_measurePreserving (i : ι) :
MeasurePreserving (cellMap i) volume volume := by
have hs : MeasurePreserving (Matrix.toLin' (shearMatrix i)) volume volume := by
refine ⟨(Matrix.toLin' (shearMatrix i)).continuous_of_finiteDimensional.measurable, ?_⟩
simpa [det_shearMatrix] using
Real.map_matrix_volume_pi_eq_smul_volume_pi
(M := shearMatrix i) (by rw [det_shearMatrix]; norm_num)
have h := (measurePreserving_add_left (volume : Measure (ι → ℝ)) (Pi.single i 1)).comp hs
simpa only [Function.comp_def, ← cellMap_eq_matrix] using h
omit [DecidableEq ι] in
theorem extrusionCell_measurable (i : ι) : MeasurableSet (extrusionCell i) := by
simp only [extrusionCell, ofPred_and, ofPred_forall]
exact ((measurableSet_le measurable_const (measurable_pi_apply i)).inter
(measurableSet_le (measurable_pi_apply i) measurable_const)).inter
(MeasurableSet.iInter fun j => MeasurableSet.iInter fun _ =>
(measurableSet_le ((measurable_pi_apply i).sub measurable_const)
(measurable_pi_apply j)).inter
(measurableSet_le (measurable_pi_apply j) (measurable_pi_apply i)))
omit [DecidableEq ι] in
theorem unitCube_volume : volume (unitCube ι) = 1 := by
simp [unitCube, Real.volume_Icc_pi]
theorem extrusionCell_volume (i : ι) : volume (extrusionCell i) = 1 := by
have hpre : cellMap i ⁻¹' extrusionCell i = unitCube ι :=
Set.ext (cellMap_mem_iff i)
rw [← (cellMap_measurePreserving i).measure_preimage
(extrusionCell_measurable i).nullMeasurableSet, hpre, unitCube_volume]
omit [DecidableEq ι] in
theorem coordinate_eq_null (i : ι) (c : ℝ) :
volume {x : ι → ℝ | x i = c} = 0 := by
simpa [volume_pi] using Measure.pi_hyperplane (fun _ : ι => (volume : Measure ℝ)) i c
theorem coordinates_eq_null (i j : ι) (hij : i ≠ j) :
volume {x : ι → ℝ | x i = x j} = 0 := by
let f : (ι → ℝ) →ₗ[ℝ] ℝ :=
(LinearMap.proj i : (ι → ℝ) →ₗ[ℝ] ℝ) - LinearMap.proj j
have hker : LinearMap.ker f ≠ ⊤ := by
intro h
have hz : f (Pi.single i 1) = 0 := by
apply LinearMap.mem_ker.mp
rw [h]
exact Submodule.mem_top
simp [f, Ne.symm hij] at hz
have hset : {x : ι → ℝ | x i = x j} = (LinearMap.ker f : Set (ι → ℝ)) := by
ext x
change x i = x j ↔ x i - x j = 0
exact sub_eq_zero.symm
rw [hset]
exact Measure.addHaar_submodule (volume : Measure (ι → ℝ)) (LinearMap.ker f) hker
omit [DecidableEq ι] in
theorem cube_cell_aedisjoint (i : ι) :
AEDisjoint volume (unitCube ι) (extrusionCell i) := by
apply measure_mono_null _ (coordinate_eq_null i 1)
rintro x ⟨hx, hi⟩
exact le_antisymm (hx.2 i) hi.1.1
theorem cells_aedisjoint : Pairwise (fun i j : ι =>
AEDisjoint volume (extrusionCell i) (extrusionCell j)) := by
intro i j hij
apply measure_mono_null _ (coordinates_eq_null i j hij)
rintro x ⟨hi, hj⟩
exact le_antisymm (hj.2 i hij).2 (hi.2 j hij.symm).2
theorem cube_add_diagonal_eq_union [Nonempty ι] :
unitCube ι + segment ℝ (0 : ι → ℝ) 1 =
unitCube ι ∪ ⋃ i, extrusionCell i := by
ext x
constructor
· rintro ⟨q, hq, z, hz, rfl⟩
change (∀ i, 0 ≤ q i) ∧ (∀ i, q i ≤ 1) at hq
rw [segment_eq_image] at hz
rcases hz with ⟨t, ht, rfl⟩
simp only [smul_zero, zero_add]
by_cases hmax : ∀ i, q i + t ≤ 1
· left
constructor <;> intro i
· simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.one_apply, mul_one]
exact add_nonneg (hq.1 i) ht.1
· simpa only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.one_apply, mul_one]
using hmax i
· right
obtain ⟨i, _, hi⟩ := Finset.exists_max_image Finset.univ q Finset.univ_nonempty
refine mem_iUnion.mpr ⟨i, ?_⟩
push Not at hmax
obtain ⟨j, hj⟩ := hmax
have hji := hi j (Finset.mem_univ j)
constructor
· constructor
· simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.one_apply, mul_one]
linarith
· simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.one_apply, mul_one]
linarith [hq.2 i, ht.2]
· intro k _
simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.one_apply, mul_one]
constructor
· linarith [hq.2 i, hq.1 k]
· linarith [hi k (Finset.mem_univ k)]
· rintro (hx | hx)
· exact ⟨x, hx, 0, left_mem_segment ℝ _ _, add_zero x⟩
· obtain ⟨i, hi⟩ := mem_iUnion.mp hx
refine ⟨fun j => x j - (x i - 1), ?_, (x i - 1) • (1 : ι → ℝ), ?_, ?_⟩
· constructor <;> intro j
· change 0 ≤ x j - (x i - 1)
by_cases hj : j = i
· subst j; linarith
· linarith [(hi.2 j hj).1]
· change x j - (x i - 1) ≤ 1
by_cases hj : j = i
· subst j; linarith
· linarith [(hi.2 j hj).2]
· rw [segment_eq_image]
refine ⟨x i - 1, ⟨by linarith [hi.1.1], by linarith [hi.1.2]⟩, ?_⟩
simp
· ext j
simp
theorem volume_cube_add_diagonal [Nonempty ι] :
volume (unitCube ι + segment ℝ (0 : ι → ℝ) 1) = Fintype.card ι + 1 := by
rw [cube_add_diagonal_eq_union,
measure_union₀ ((MeasurableSet.iUnion extrusionCell_measurable).nullMeasurableSet)
(AEDisjoint.iUnion_right_iff.mpr cube_cell_aedisjoint),
measure_iUnion₀ cells_aedisjoint (fun i => (extrusionCell_measurable i).nullMeasurableSet),
unitCube_volume]
simp [extrusionCell_volume, add_comm]
theorem volume_Icc_add_segment_one (d : ℕ) (hd : 0 < d) :
volume (Icc (0 : Fin d → ℝ) 1 + segment ℝ (0 : Fin d → ℝ) 1) = d + 1 := by
let : Nonempty (Fin d) := ⟨⟨0, hd⟩⟩
simpa [unitCube] using (volume_cube_add_diagonal (ι := Fin d))
end Paper092
end OAI
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