| import Mathlib |
| import OAI.Geometry.Borsuk.Support.Chambers |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| section ChamberPath |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| noncomputable def chamberPath (c u : E) (t : ℝ) : E := |
| ‖(1-t) • u + t • c‖⁻¹ • ((1-t) • u + t • c) |
|
|
| lemma signChamber_smul {Z : Set E} {c u : E} (hu : u ∈ signChamber Z c) |
| {t : ℝ} (ht : 0 < t) : t • u ∈ signChamber Z c := by |
| intro z hz |
| rw [real_inner_smul_left,mul_left_comm] |
| exact mul_pos ht (hu z hz) |
|
|
| lemma chamber_interpolation_mem {Z : Set E} {c u : E} (hc : c ∈ safeNormals Z) |
| (hu : u ∈ signChamber Z c) {t : ℝ} (ht : t ∈ Set.Icc 0 1) : |
| (1-t) • u + t • c ∈ signChamber Z c := |
| signChamber_convex Z c hu (mem_signChamber_self hc) (sub_nonneg.mpr ht.2) ht.1 (by ring) |
|
|
| lemma chamber_interpolation_ne_zero {Z : Set E} {c u : E} (hcZ : c ∈ Z) |
| (hc : c ∈ safeNormals Z) (hu : u ∈ signChamber Z c) |
| {t : ℝ} (ht : t ∈ Set.Icc 0 1) : (1-t) • u + t • c ≠ 0 := by |
| intro he |
| have h := chamber_interpolation_mem hc hu ht c hcZ |
| simp [he] at h |
|
|
| lemma chamberPath_norm {Z : Set E} {c u : E} (hcZ : c ∈ Z) |
| (hc : c ∈ safeNormals Z) (hu : u ∈ signChamber Z c) |
| {t : ℝ} (ht : t ∈ Set.Icc 0 1) : ‖chamberPath c u t‖ = 1 := by |
| have hn := chamber_interpolation_ne_zero hcZ hc hu ht |
| simp only [chamberPath,norm_smul,norm_inv,norm_norm,inv_mul_cancel₀ (norm_ne_zero_iff.mpr hn)] |
|
|
| lemma chamberPath_mem {Z : Set E} {c u : E} (hcZ : c ∈ Z) |
| (hc : c ∈ safeNormals Z) (hu : u ∈ signChamber Z c) |
| {t : ℝ} (ht : t ∈ Set.Icc 0 1) : chamberPath c u t ∈ signChamber Z c := |
| signChamber_smul (chamber_interpolation_mem hc hu ht) |
| (inv_pos.mpr (norm_pos_iff.mpr (chamber_interpolation_ne_zero hcZ hc hu ht))) |
|
|
| lemma chamberPath_zero (c u : E) (hu : ‖u‖ = 1) : chamberPath c u 0 = u := by |
| simp [chamberPath,hu] |
|
|
| lemma chamberPath_one (c u : E) (hc : ‖c‖ = 1) : chamberPath c u 1 = c := by |
| simp [chamberPath,hc] |
|
|
| lemma chamber_add_ne_zero {Z : Set E} {c u : E} (hcZ : c ∈ Z) |
| (hc : c ∈ safeNormals Z) (hu : u ∈ signChamber Z c) : c+u ≠ 0 := by |
| intro he |
| have h := hu c hcZ |
| have hn := hc c hcZ |
| have hei : ⟪c,c⟫ + ⟪u,c⟫ = 0 := by rw [← inner_add_left,he,inner_zero_left] |
| have hei' : ⟪u,c⟫ = -⟪c,c⟫ := by linarith |
| rw [hei'] at h |
| nlinarith [sq_nonneg ⟪c,c⟫] |
|
|
| lemma chamberPath_continuousOn {Z : Set E} {c : E} (hcZ : c ∈ Z) |
| (hc : c ∈ safeNormals Z) : ContinuousOn (fun p : ℝ × E => chamberPath c p.2 p.1) |
| (Set.Icc (0:ℝ) 1 ×ˢ signChamber Z c) := by |
| have hs : Continuous (fun p : ℝ × E => (1-p.1) • p.2 + p.1 • c) := |
| ((continuous_const.sub continuous_fst).smul continuous_snd).add |
| (continuous_fst.smul continuous_const) |
| exact (hs.norm.continuousOn.inv₀ (fun p hp => norm_ne_zero_iff.mpr |
| (chamber_interpolation_ne_zero hcZ hc hp.2 hp.1))).smul hs.continuousOn |
|
|
| end ChamberPath |
| end BorsukNine |
|
|
| end OAI |
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|
