| import Mathlib |
| import OAI.Geometry.Borsuk.Projectors.ChartSurgery |
| import OAI.Geometry.Borsuk.MatrixGeometry.CenterApproximation |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| open Set |
| section Purity |
| variable {k : ℕ} {ι : Type*} [Fintype ι] [DecidableEq ι] |
| local notation "E" => EuclideanSpace ℝ (Fin (k+1)) |
|
|
| lemma Admissible.maximal_support_complement_bound {f : E → ι → ℝ} (hf : Admissible f) |
| (hdim : Fintype.card (UpperTriangle (Fin (k+1))) = Fintype.card ι) |
| (hn : 1 < Fintype.card ι) {w : E} (hw : ‖w‖ = 1) |
| (hmax : ∀ u : E, ‖u‖ = 1 → positiveSupport f w ⊆ positiveSupport f u → |
| positiveSupport f u = positiveSupport f w) : |
| Fintype.card ↥((positiveSupport f w)ᶜ : Finset ι) ≤ Fintype.card (UpperTriangle (Fin k)) := by |
| classical |
| by_contra hb |
| have hd : Fintype.card (UpperTriangle (Fin k)) < |
| Fintype.card ↥((positiveSupport f w)ᶜ : Finset ι) := lt_of_not_ge hb |
| let I := positiveSupport f w |
| have hI : I.Nonempty := hf.support_nonempty hw |
| have hr : ∀ i ∈ I, 0 < f w i := fun i hi => (mem_positiveSupport f w i).mp hi |
| have hro : ∀ i ∉ I, f w i = 0 := by |
| intro i hi |
| exact le_antisymm (not_lt.mp (fun hp => hi ((mem_positiveSupport f w i).mpr hp))) (hf.nonneg w hw i) |
| obtain ⟨C,hCK,hCD,hcover⟩ := compact_disjoint_chambers (hf.orientedLevel_compact w (f w)) |
| (fun u hu => hf.commonPositiveCone_interior_safe I (orientedLevel_cone hr hu)) |
| let c : ↥C → E := Subtype.val |
| have hc : ∀ j, c j ∈ commonPositiveCone f I := fun j => orientedLevel_cone hr (hCK j.property) |
| have hcn : ∀ j, ‖c j‖ = 1 := fun j => (hCK j.property).1 |
| obtain ⟨z,hz,hzs,g,hg,hgs,hge,hgm⟩ := hf.finite_center_approximations hd c |
| (fun j => hf.commonPositiveCone_safe I (hc j)) hcn |
| have hj : 0 < Fintype.card ↥(Iᶜ : Finset ι) := lt_of_le_of_lt (Nat.zero_le _) hd |
| let : Nonempty ↥(Iᶜ : Finset ι) := Fintype.card_pos_iff.mp hj |
| have hε : 0 < (2*(Fintype.card ↥(Iᶜ : Finset ι):ℝ))⁻¹ := by positivity |
| exact hf.chart_surgery_impossible hdim hn hI hmax hw hr hr hro (hf.sum_one w hw) |
| c hc hcn hCD hcover hε g hg hgs hge z hz hzs hgm |
|
|
| end Purity |
| end BorsukNine |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| section PureFacets |
| variable {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [Fintype ι] [DecidableEq ι] |
| omit [InnerProductSpace ℝ E] [DecidableEq ι] in |
| lemma exists_maximal_support (f : E → ι → ℝ) {w : E} (hw : ‖w‖ = 1) : |
| ∃ u, ‖u‖ = 1 ∧ positiveSupport f w ⊆ positiveSupport f u ∧ |
| ∀ v, ‖v‖ = 1 → positiveSupport f u ⊆ positiveSupport f v → |
| positiveSupport f v = positiveSupport f u := by |
| classical |
| let S : Set (Finset ι) := {s | ∃ u, ‖u‖ = 1 ∧ positiveSupport f w ⊆ s ∧ positiveSupport f u = s} |
| obtain ⟨s,hs,hm⟩ := (Set.toFinite S).exists_maximal (show S.Nonempty from ⟨_,w,hw,Finset.Subset.rfl,rfl⟩) |
| obtain ⟨u,hu,hwu,rfl⟩ := hs |
| refine ⟨u,hu,hwu,?_⟩ |
| intro v hv huv |
| exact le_antisymm (hm ⟨v,hv,hwu.trans huv,rfl⟩ huv) huv |
| end PureFacets |
|
|
| lemma Admissible.maximal_support_card_four {f : Vector4 → Fin 10 → ℝ} (hf : Admissible f) |
| {w : Vector4} (hw : ‖w‖ = 1) |
| (hm : ∀ u, ‖u‖ = 1 → positiveSupport f w ⊆ positiveSupport f u → |
| positiveSupport f u = positiveSupport f w) : (positiveSupport f w).card = 4 := by |
| have hl := hf.support_card_four_le hw |
| have hb := hf.maximal_support_complement_bound (k := 3) (by decide) (by decide) hw hm |
| have hc : Fintype.card (UpperTriangle (Fin 3)) = 6 := by decide |
| rw [Fintype.card_coe,Finset.card_compl,Fintype.card_fin,hc] at hb |
| omega |
|
|
| lemma Admissible.maximal_support_card_three {f : EuclideanSpace ℝ (Fin 3) → Fin 6 → ℝ} |
| (hf : Admissible f) {w : EuclideanSpace ℝ (Fin 3)} (hw : ‖w‖ = 1) |
| (hm : ∀ u, ‖u‖ = 1 → positiveSupport f w ⊆ positiveSupport f u → |
| positiveSupport f u = positiveSupport f w) : (positiveSupport f w).card = 3 := by |
| have hl := hf.support_card_three_le hw |
| have hb := hf.maximal_support_complement_bound (k := 2) (by decide) (by decide) hw hm |
| have hc : Fintype.card (UpperTriangle (Fin 2)) = 3 := by decide |
| rw [Fintype.card_coe,Finset.card_compl,Fintype.card_fin,hc] at hb |
| omega |
|
|
| lemma Admissible.support_extends_four {f : Vector4 → Fin 10 → ℝ} (hf : Admissible f) |
| {w : Vector4} (hw : ‖w‖ = 1) : |
| ∃ u, ‖u‖ = 1 ∧ positiveSupport f w ⊆ positiveSupport f u ∧ (positiveSupport f u).card = 4 := by |
| obtain ⟨u,hu,hwu,hm⟩ := exists_maximal_support f hw |
| exact ⟨u,hu,hwu,hf.maximal_support_card_four hu hm⟩ |
|
|
| lemma Admissible.support_extends_three {f : EuclideanSpace ℝ (Fin 3) → Fin 6 → ℝ} |
| (hf : Admissible f) {w : EuclideanSpace ℝ (Fin 3)} (hw : ‖w‖ = 1) : |
| ∃ u, ‖u‖ = 1 ∧ positiveSupport f w ⊆ positiveSupport f u ∧ (positiveSupport f u).card = 3 := by |
| obtain ⟨u,hu,hwu,hm⟩ := exists_maximal_support f hw |
| exact ⟨u,hu,hwu,hf.maximal_support_card_three hu hm⟩ |
| end BorsukNine |
|
|
| open Set MeasureTheory |
|
|
| end OAI |
|
|
