| import Mathlib |
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| namespace OAI |
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| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| section Chambers |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
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| def safeNormals (Z : Set E) : Set E := {x | ∀ z ∈ Z, ⟪x,z⟫ ≠ 0} |
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| def signChamber (Z : Set E) (c : E) : Set E := {x | ∀ z ∈ Z, 0 < ⟪c,z⟫ * ⟪x,z⟫} |
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| lemma mem_signChamber_self {Z : Set E} {c : E} (hc : c ∈ safeNormals Z) : |
| c ∈ signChamber Z c := by |
| intro z hz |
| exact mul_self_pos.mpr (hc z hz) |
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| lemma signChamber_subset_safeNormals (Z : Set E) (c : E) : |
| signChamber Z c ⊆ safeNormals Z := by |
| intro x hx z hz h |
| simpa [h] using hx z hz |
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| lemma signChamber_convex (Z : Set E) (c : E) : Convex ℝ (signChamber Z c) := by |
| intro x hx y hy a b ha hb hab z hz |
| simp only [inner_add_left,real_inner_smul_left,mul_add] |
| have h1 := hx z hz |
| have h2 := hy z hz |
| have hn1 := mul_nonneg ha h1.le |
| have hn2 := mul_nonneg hb h2.le |
| have he : ⟪c,z⟫ * (a * ⟪x,z⟫) + ⟪c,z⟫ * (b * ⟪y,z⟫) = |
| a * (⟪c,z⟫ * ⟪x,z⟫) + b * (⟪c,z⟫ * ⟪y,z⟫) := by ring |
| rw [he] |
| rcases lt_or_eq_of_le ha with ha' | ha' |
| · exact add_pos_of_pos_of_nonneg (mul_pos ha' h1) hn2 |
| · have hb' : b = 1 := by linarith |
| simp [← ha',hb',h2] |
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| lemma signChamber_eq_of_mem {Z : Set E} {c d x : E} |
| (hc : x ∈ signChamber Z c) (hd : x ∈ signChamber Z d) : |
| signChamber Z c = signChamber Z d := by |
| ext y |
| constructor <;> intro hy z hz |
| · have h1 := hc z hz |
| have h2 := hd z hz |
| have h3 := hy z hz |
| rcases mul_pos_iff.mp h1 with h1 | h1 <;> |
| rcases mul_pos_iff.mp h2 with h2 | h2 <;> |
| rcases mul_pos_iff.mp h3 with h3 | h3 |
| all_goals first | exact mul_pos h2.1 h3.2 | exact mul_pos_of_neg_of_neg h2.1 h3.2 | linarith [h1.1,h1.2,h2.1,h2.2,h3.1,h3.2] |
| · have h1 := hc z hz |
| have h2 := hd z hz |
| have h3 := hy z hz |
| rcases mul_pos_iff.mp h1 with h1 | h1 <;> |
| rcases mul_pos_iff.mp h2 with h2 | h2 <;> |
| rcases mul_pos_iff.mp h3 with h3 | h3 |
| all_goals first | exact mul_pos h1.1 h3.2 | exact mul_pos_of_neg_of_neg h1.1 h3.2 | linarith [h1.1,h1.2,h2.1,h2.2,h3.1,h3.2] |
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| lemma signChamber_eq_or_disjoint (Z : Set E) (c d : E) : |
| signChamber Z c = signChamber Z d ∨ Disjoint (signChamber Z c) (signChamber Z d) := by |
| by_cases h : Disjoint (signChamber Z c) (signChamber Z d) |
| · exact Or.inr h |
| · rw [Set.disjoint_left] at h |
| push Not at h |
| obtain ⟨x,hc,hd⟩ := h |
| exact Or.inl (signChamber_eq_of_mem hc hd) |
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| lemma interior_safe_mem_chamber {Z : Set E} {c x : E} |
| (hx : x ∈ interior (safeNormals Z)) (hxc : x ∈ signChamber Z c) : |
| x ∈ interior (signChamber Z c) := by |
| obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp hx) |
| apply mem_interior_iff_mem_nhds.mpr |
| apply Filter.mem_of_superset (Metric.ball_mem_nhds x hε) |
| intro y hy z hz |
| have hc := hxc z hz |
| have hn : ∀ w ∈ Metric.ball x ε, ⟪w,z⟫ ≠ 0 := fun w hw => hball hw z hz |
| have hf : ContinuousOn (fun w : E => ⟪w,z⟫) (Metric.ball x ε) := |
| (continuous_id.inner continuous_const).continuousOn |
| have hxball : x ∈ Metric.ball x ε := Metric.mem_ball_self hε |
| rcases mul_pos_iff.mp hc with hc | hc |
| · exact mul_pos hc.1 ((convex_ball x ε).isPreconnected.lt_of_ne hf hn ⟨x,hxball,hc.2⟩ hy) |
| · exact mul_pos_of_neg_of_neg hc.1 ((convex_ball x ε).isPreconnected.gt_of_ne hf hn ⟨x,hxball,hc.2⟩ hy) |
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| lemma interior_safe_mem_own_chamber {Z : Set E} {c : E} |
| (hc : c ∈ interior (safeNormals Z)) : c ∈ interior (signChamber Z c) := |
| interior_safe_mem_chamber hc (mem_signChamber_self (interior_subset hc)) |
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| lemma signChamber_interior_pairwise (Z : Set E) : |
| Pairwise (fun U W : {U : Set E // ∃ c, U = interior (signChamber Z c)} => |
| Disjoint (U.val) (W.val)) := by |
| intro U W hne |
| obtain ⟨c,hc⟩ := U.property |
| obtain ⟨d,hd⟩ := W.property |
| rcases signChamber_eq_or_disjoint Z c d with he | hdj |
| · exact False.elim (hne (Subtype.ext (hc.trans ((congrArg interior he).trans hd.symm)))) |
| · rw [hc,hd] |
| exact hdj.mono interior_subset interior_subset |
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| lemma compact_finite_chambers {Z K : Set E} (hK : IsCompact K) |
| (hsub : K ⊆ interior (safeNormals Z)) : |
| ∃ C : Finset E, (↑C : Set E) ⊆ K ∧ K ⊆ ⋃ c ∈ (↑C : Set E), interior (signChamber Z c) := by |
| classical |
| obtain ⟨C,hCK,hCf,hcover⟩ := hK.elim_finite_subcover_image |
| (b := K) (c := fun c : E => interior (signChamber Z c)) (fun _ _ => isOpen_interior) |
| (fun c hc => Set.mem_iUnion₂.mpr ⟨c,hc,interior_safe_mem_own_chamber (hsub hc)⟩) |
| exact ⟨hCf.toFinset,by simpa using hCK,by simpa using hcover⟩ |
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| end Chambers |
| end BorsukNine |
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| end OAI |
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