| import Mathlib |
| import OAI.Geometry.Borsuk.Projectors.Tangent |
| import OAI.Geometry.Borsuk.Counterexample |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| namespace BorsukNine |
|
|
| abbrev Vector9 := EuclideanSpace ℝ (Fin 9) |
|
|
| noncomputable def coordinates9 (A : Matrix4) : Vector9 := |
| WithLp.toLp 2 ![(A (0, 0) - A (1, 1)) / Real.sqrt 2, |
| (A (0, 0) + A (1, 1) - 2 * A (2, 2)) / Real.sqrt 6, |
| (A (0, 0) + A (1, 1) + A (2, 2) - 3 * A (3, 3)) / Real.sqrt 12, |
| Real.sqrt 2 * A (0, 1), Real.sqrt 2 * A (0, 2), Real.sqrt 2 * A (0, 3), |
| Real.sqrt 2 * A (1, 2), Real.sqrt 2 * A (1, 3), Real.sqrt 2 * A (2, 3)] |
|
|
| theorem coordinates9_sub (A B : Matrix4) : |
| coordinates9 (A - B) = coordinates9 A - coordinates9 B := by |
| ext i |
| fin_cases i <;> simp [coordinates9] <;> ring |
|
|
| theorem coordinates9_norm_sq (A : Matrix4) |
| (hA : ∀ i j : Fin 4, A (i, j) = A (j, i)) |
| (htr : ∑ i : Fin 4, A (i, i) = 0) : |
| ‖coordinates9 A‖ ^ 2 = ‖A‖ ^ 2 := by |
| have h2 : (Real.sqrt 2) ^ 2 = (2 : ℝ) := Real.sq_sqrt (by norm_num) |
| have h6 : (Real.sqrt 6) ^ 2 = (6 : ℝ) := Real.sq_sqrt (by norm_num) |
| have h12 : (Real.sqrt 12) ^ 2 = (12 : ℝ) := Real.sq_sqrt (by norm_num) |
| have h33 : A (3, 3) = -A (0, 0) - A (1, 1) - A (2, 2) := by |
| norm_num [Fin.sum_univ_succ] at htr |
| change A (0, 0) + (A (1, 1) + (A (2, 2) + A (3, 3))) = 0 at htr |
| linarith |
| rw [EuclideanSpace.real_norm_sq_eq, EuclideanSpace.real_norm_sq_eq] |
| simp [coordinates9, Fintype.sum_prod_type, Fin.sum_univ_succ, div_pow, mul_pow, |
| h2, h6, h12, hA 1 0, hA 2 0, hA 3 0, hA 2 1, hA 3 1, hA 3 2, h33] |
| ring |
|
|
| theorem coordinates9_dist (A B : Matrix4) |
| (hA : A ∈ traceOneSymmetric) (hB : B ∈ traceOneSymmetric) : |
| dist (coordinates9 A) (coordinates9 B) = dist A B := by |
| have hs : ∀ i j : Fin 4, (A - B) (i, j) = (A - B) (j, i) := by |
| intro i j |
| change A (i, j) - B (i, j) = A (j, i) - B (j, i) |
| rw [hA.1 i j, hB.1 i j] |
| have ht : ∑ i : Fin 4, (A - B) (i, i) = 0 := by |
| change (∑ i : Fin 4, (A (i, i) - B (i, i))) = 0 |
| rw [Finset.sum_sub_distrib, hA.2, hB.2, sub_self] |
| have hn := coordinates9_norm_sq (A - B) hs ht |
| rw [coordinates9_sub] at hn |
| rw [dist_eq_norm, dist_eq_norm] |
| nlinarith [norm_nonneg (coordinates9 A - coordinates9 B), norm_nonneg (A - B)] |
|
|
| theorem coordinates9_isometry : |
| Isometry (fun A : traceOneSymmetric => coordinates9 A.val) := |
| Isometry.of_dist_eq fun A B => coordinates9_dist A B A.property B.property |
|
|
| theorem continuous_coordinates9 : Continuous coordinates9 := by |
| apply (PiLp.continuous_toLp 2 _).comp |
| apply continuous_pi |
| intro i |
| fin_cases i <;> simp <;> fun_prop |
|
|
| theorem coordinates9_diam (S : Set Matrix4) (hS : S ⊆ traceOneSymmetric) : |
| Metric.diam (coordinates9 '' S) = Metric.diam S := by |
| let f : S → Vector9 := fun A => coordinates9 A.val |
| have hf : Isometry f := Isometry.of_dist_eq fun A B => |
| coordinates9_dist A B (hS A.property) (hS B.property) |
| have hr : Set.range f = coordinates9 '' S := by |
| ext x |
| constructor |
| · rintro ⟨A, rfl⟩ |
| exact ⟨A.val, A.property, rfl⟩ |
| · rintro ⟨A, hA, rfl⟩ |
| exact ⟨⟨A, hA⟩, rfl⟩ |
| have hi : Isometry (fun A : S => (A : Matrix4)) := isometry_subtype_coe |
| have hs : Set.range (fun A : S => (A : Matrix4)) = S := by |
| ext x |
| constructor |
| · rintro ⟨A, rfl⟩ |
| exact A.property |
| · intro hx |
| exact ⟨⟨x, hx⟩, rfl⟩ |
| calc |
| Metric.diam (coordinates9 '' S) = Metric.diam (Set.range f) := by rw [hr] |
| _ = Metric.diam (Set.univ : Set S) := hf.diam_range |
| _ = Metric.diam (Set.range (fun A : S => (A : Matrix4))) := hi.diam_range.symm |
| _ = Metric.diam S := by rw [hs] |
|
|
| theorem coordinates9_cover_pullback (S : Set Matrix4) (hS : S ⊆ traceOneSymmetric) |
| (C : Fin 10 → Set Vector9) (hC : ∀ i, C i ⊆ coordinates9 '' S) |
| (hcover : coordinates9 '' S ⊆ ⋃ i, C i) |
| (hdiam : ∀ i, Metric.diam (C i) < Real.sqrt 2) : |
| ∃ D : Fin 10 → Set Matrix4, |
| (∀ i, D i ⊆ S) ∧ S ⊆ ⋃ i, D i ∧ |
| ∀ i, Metric.diam (D i) < Real.sqrt 2 := by |
| let D : Fin 10 → Set Matrix4 := fun i => {A | A ∈ S ∧ coordinates9 A ∈ C i} |
| have him : ∀ i, coordinates9 '' D i = C i := by |
| intro i |
| ext x |
| constructor |
| · rintro ⟨A, hA, rfl⟩ |
| exact hA.2 |
| · intro hx |
| obtain ⟨A, hA, rfl⟩ := hC i hx |
| exact ⟨A, ⟨hA, hx⟩, rfl⟩ |
| refine ⟨D, fun _ _ h => h.1, ?_, ?_⟩ |
| · intro A hA |
| obtain ⟨i, hi⟩ := Set.mem_iUnion.mp (hcover ⟨A, hA, rfl⟩) |
| exact Set.mem_iUnion.mpr ⟨i, hA, hi⟩ |
| · intro i |
| have hD : D i ⊆ traceOneSymmetric := fun _ h => hS h.1 |
| rw [← coordinates9_diam (D i) hD, him i] |
| exact hdiam i |
|
|
| theorem euclidean_nine_counterexample : |
| ∃ X : Set (EuclideanSpace ℝ (Fin 9)), |
| IsCompact X ∧ Metric.diam X = Real.sqrt 2 ∧ |
| ¬ (∃ C : Fin 10 → Set (EuclideanSpace ℝ (Fin 9)), |
| (∀ i, C i ⊆ X) ∧ X ⊆ ⋃ i, C i ∧ |
| ∀ i, Metric.diam (C i) < Real.sqrt 2) := by |
| obtain ⟨hcompact, htrace, hdiam, hnocover⟩ := main_theorem |
| refine ⟨coordinates9 '' projectorSet, hcompact.image continuous_coordinates9, ?_, ?_⟩ |
| · rw [coordinates9_diam projectorSet htrace, hdiam] |
| · rintro ⟨C, hC, hcover, hsmall⟩ |
| exact hnocover (coordinates9_cover_pullback projectorSet htrace C hC hcover hsmall) |
|
|
| end BorsukNine |
|
|
| end OAI |
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|
