| import Mathlib |
| import OAI.Geometry.Borsuk.Projectors.Restrict |
| import OAI.Geometry.Borsuk.Support.Chambers |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| section ActiveChambers |
| variable {E W ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [NormedAddCommGroup W] [InnerProductSpace ℝ W] [Fintype ι] |
|
|
| def activeVectors (f : E → ι → ℝ) (I : Finset ι) : Set E := |
| {z | ∃ i ∈ I, 0 < homogeneousLift f z i} |
|
|
| def commonPositiveCone (f : E → ι → ℝ) (I : Finset ι) : Set E := |
| {z | ∀ i ∈ I, 0 < homogeneousLift f z i} |
|
|
| lemma Admissible.commonPositiveCone_isOpen {f : E → ι → ℝ} (hf : Admissible f) |
| (I : Finset ι) : IsOpen (commonPositiveCone f I) := by |
| have he : commonPositiveCone f I = |
| ⋂ i ∈ (I : Set ι), {x | 0 < homogeneousLift f x i} := by ext x; simp [commonPositiveCone] |
| rw [he] |
| exact isOpen_biInter_finset (fun i _ => isOpen_lt continuous_const |
| ((continuous_apply i).comp hf.lift_continuous)) |
|
|
| lemma Admissible.commonPositiveCone_safe {f : E → ι → ℝ} (hf : Admissible f) |
| (I : Finset ι) : commonPositiveCone f I ⊆ safeNormals (activeVectors f I) := by |
| intro x hx z hz horth |
| obtain ⟨i,hi,hzi⟩ := hz |
| have hxz := mul_pos (hx i hi) hzi |
| rw [hf.lift_orthogonal horth i] at hxz |
| exact (lt_irrefl 0) hxz |
|
|
| lemma Admissible.commonPositiveCone_interior_safe {f : E → ι → ℝ} |
| (hf : Admissible f) (I : Finset ι) : |
| commonPositiveCone f I ⊆ interior (safeNormals (activeVectors f I)) := |
| interior_maximal (hf.commonPositiveCone_safe I) (hf.commonPositiveCone_isOpen I) |
|
|
| omit [Fintype ι] in |
| lemma commonPositiveCone_active {f : E → ι → ℝ} {I : Finset ι} (hI : I.Nonempty) |
| {c : E} (hc : c ∈ commonPositiveCone f I) : c ∈ activeVectors f I := by |
| obtain ⟨i,hi⟩ := hI |
| exact ⟨i,hi,hc i hi⟩ |
|
|
| omit [Fintype ι] in |
| lemma commonPositiveCone_ne_zero {f : E → ι → ℝ} {I : Finset ι} (hI : I.Nonempty) |
| {c : E} (hc : c ∈ commonPositiveCone f I) : c ≠ 0 := by |
| intro he |
| obtain ⟨i,hi⟩ := hI |
| have hh := hc i hi |
| simp [he] at hh |
|
|
| omit [Fintype ι] in |
| lemma activeChamber_inner_pos {f : E → ι → ℝ} {I : Finset ι} (hI : I.Nonempty) |
| {c x : E} (hc : c ∈ commonPositiveCone f I) |
| (hx : x ∈ signChamber (activeVectors f I) c) : 0 < ⟪c,x⟫ := by |
| have hh := hx c (commonPositiveCone_active hI hc) |
| rw [real_inner_comm c x] at hh |
| exact (mul_pos_iff_of_pos_left (real_inner_self_pos.mpr |
| (commonPositiveCone_ne_zero hI hc))).mp hh |
|
|
| lemma Admissible.safe_restriction_zero {f : E → ι → ℝ} (hf : Admissible f) |
| {I : Finset ι} {x : E} (hx : x ∈ safeNormals (activeVectors f I)) |
| (e : W →ₗᵢ[ℝ] E) (he : ∀ u, ⟪x,e u⟫ = 0) |
| {u : W} (hu : ‖u‖ = 1) {i : ι} (hi : i ∈ I) : f (e u) i = 0 := by |
| have hunit : ‖e u‖ = 1 := by simpa using hu |
| apply le_antisymm _ (hf.nonneg _ hunit i) |
| apply le_of_not_gt |
| intro hp |
| have hz : e u ∈ activeVectors f I := ⟨i,hi,by simpa [homogeneousLift_unit _ hunit] using hp⟩ |
| exact hx (e u) hz (he u) |
|
|
| lemma Admissible.safe_restriction [DecidableEq ι] {f : E → ι → ℝ} (hf : Admissible f) |
| {I : Finset ι} {x : E} (hx : x ∈ safeNormals (activeVectors f I)) |
| (e : W →ₗᵢ[ℝ] E) (he : ∀ u, ⟪x,e u⟫ = 0) : |
| Admissible (fun u (i : ↥(Iᶜ : Finset ι)) => f (e u) i.val) := by |
| classical |
| apply (hf.comp_isometry e).restrict_labels Iᶜ |
| intro u hu i hi |
| exact hf.safe_restriction_zero hx e he hu (by simpa using hi) |
|
|
| end ActiveChambers |
| end BorsukNine |
|
|
| end OAI |
|
|
