| import Mathlib |
| import OAI.Geometry.Borsuk.Simplicial.OddSphereDimensionBoundFintype |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| section Restrict |
| variable {V W ι : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [NormedAddCommGroup W] [InnerProductSpace ℝ W] [Fintype ι] |
|
|
| lemma Admissible.comp_isometry {f : V → ι → ℝ} (hf : Admissible f) (e : W →ₗᵢ[ℝ] V) : |
| Admissible (fun x => f (e x)) := by |
| refine ⟨fun i => (hf.smooth i).comp e.contDiff,?_,?_,?_,?_⟩ |
| · intro x i |
| rw [map_neg,hf.antipodal] |
| · intro x hx i |
| exact hf.nonneg _ ((e.norm_map x).trans hx) i |
| · intro x hx |
| exact hf.sum_one _ ((e.norm_map x).trans hx) |
| · intro x y hx hy hxy i |
| exact hf.orthogonal _ _ ((e.norm_map x).trans hx) ((e.norm_map y).trans hy) |
| ((e.inner_map_map x y).trans hxy) i |
|
|
| lemma Admissible.restrict_labels {f : V → ι → ℝ} (hf : Admissible f) |
| (s : Finset ι) (hzero : ∀ x, ‖x‖ = 1 → ∀ i ∉ s, f x i = 0) : |
| Admissible (fun x (i : s) => f x i.val) := by |
| classical |
| refine ⟨fun i => hf.smooth i.val,fun x i => hf.antipodal x i.val, |
| fun x hx i => hf.nonneg x hx i.val,?_,fun x y hx hy hxy i => hf.orthogonal x y hx hy hxy i.val⟩ |
| intro x hx |
| rw [← hf.sum_one x hx] |
| rw [Finset.sum_coe_sort] |
| exact Finset.sum_subset (Finset.subset_univ s) (fun i _ hi => hzero x hx i hi) |
|
|
| end Restrict |
|
|
| lemma orthogonal_isometry (k : ℕ) {u : EuclideanSpace ℝ (Fin (k+1))} (hu : ‖u‖ = 1) : |
| ∃ e : EuclideanSpace ℝ (Fin k) →ₗᵢ[ℝ] EuclideanSpace ℝ (Fin (k+1)), |
| ∀ x, ⟪u,e x⟫ = 0 := by |
| let W := (Submodule.span ℝ {u})ᗮ |
| have hu0 : u ≠ 0 := by intro h; simp [h] at hu |
| have hdim : Module.finrank ℝ W = k := by |
| let : Fact (Module.finrank ℝ (EuclideanSpace ℝ (Fin (k+1))) = k+1) := ⟨by simp⟩ |
| exact Submodule.finrank_orthogonal_span_singleton hu0 |
| let b := (stdOrthonormalBasis ℝ W).reindex (finCongr hdim) |
| refine ⟨W.subtypeₗᵢ.comp b.repr.symm.toLinearIsometry,?_⟩ |
| intro x |
| exact (Submodule.mem_orthogonal_singleton_iff_inner_right.mp (b.repr.symm x).property) |
|
|
| lemma Admissible.support_card_four_le {f : Vector4 → Fin 10 → ℝ} (hf : Admissible f) |
| {u : Vector4} (hu : ‖u‖ = 1) : (positiveSupport f u).card ≤ 4 := by |
| classical |
| obtain ⟨e,he⟩ := orthogonal_isometry 3 hu |
| let s := (positiveSupport f u)ᶜ |
| have hzero : ∀ x, ‖x‖ = 1 → ∀ i ∉ s, f (e x) i = 0 := by |
| intro x hx i hi |
| have hip : 0 < f u i := (mem_positiveSupport _ _ _).mp (by simpa [s] using hi) |
| have hn := hf.nonneg (e x) (by simpa using hx) i |
| exact le_antisymm (le_of_not_gt (fun hp => hf.orthogonal u (e x) hu |
| (by simpa using hx) (he x) i ⟨hip,hp⟩)) hn |
| have hb := ((hf.comp_isometry e).restrict_labels s hzero).label_bound |
| have hc : Fintype.card (UpperTriangle (Fin 3)) = 6 := by decide |
| rw [hc,Fintype.card_coe,Finset.card_compl] at hb |
| rw [Fintype.card_fin] at hb |
| omega |
|
|
| lemma Admissible.support_card_three_le {f : EuclideanSpace ℝ (Fin 3) → Fin 6 → ℝ} |
| (hf : Admissible f) {u : EuclideanSpace ℝ (Fin 3)} (hu : ‖u‖ = 1) : |
| (positiveSupport f u).card ≤ 3 := by |
| classical |
| obtain ⟨e,he⟩ := orthogonal_isometry 2 hu |
| let s := (positiveSupport f u)ᶜ |
| have hzero : ∀ x, ‖x‖ = 1 → ∀ i ∉ s, f (e x) i = 0 := by |
| intro x hx i hi |
| have hip : 0 < f u i := (mem_positiveSupport _ _ _).mp (by simpa [s] using hi) |
| have hn := hf.nonneg (e x) (by simpa using hx) i |
| exact le_antisymm (le_of_not_gt (fun hp => hf.orthogonal u (e x) hu |
| (by simpa using hx) (he x) i ⟨hip,hp⟩)) hn |
| have hb := ((hf.comp_isometry e).restrict_labels s hzero).label_bound |
| have hc : Fintype.card (UpperTriangle (Fin 2)) = 3 := by decide |
| rw [hc,Fintype.card_coe,Finset.card_compl] at hb |
| rw [Fintype.card_fin] at hb |
| omega |
|
|
| end BorsukNine |
|
|
| end OAI |
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|
