| import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar |
| 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 |
|
|
