| import OAI.Geometry.CylinderCovering.Rational.Wedge |
| import OAI.Geometry.CylinderCovering.Convexity.RelaxedInversion |
| import OAI.Geometry.CylinderCovering.Rational.Lipschitz |
| import OAI.Geometry.CylinderCovering.Rational.GlobalInversionDefinitions |
|
|
| namespace OAI |
|
|
| open Set |
|
|
| namespace SlopeRational |
|
|
| theorem rational_wedge_isClosed : IsClosed wedge := by |
| apply isClosed_le |
| · fun_prop |
| · unfold radius; fun_prop |
|
|
| theorem rational_wedge_convex : Convex ℝ wedge := by |
| intro p hp q hq u v hu hv huv |
| rw [mem_wedge_iff_walls] at hp hq ⊢ |
| simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul] |
| constructor |
| · have h₁ := mul_nonneg hu hp.1 |
| have h₂ := mul_nonneg hv hq.1 |
| unfold radius at * |
| nlinarith |
| · have h₁ := mul_nonneg hu hp.2 |
| have h₂ := mul_nonneg hv hq.2 |
| unfold radius at * |
| nlinarith |
|
|
| theorem global_wall_coefficients {d : ℝ} (hd : 0 < d) {p : Plane} (hp : p ∈ wedge) : |
| |(2 * p 0 + 2 * d + radius (p 1)) / (8 * (radius (p 1) + d) ^ 2)| ≤ 1 / (4 * d) ∧ |
| |(-2 * p 0 + 2 * d + radius (p 1)) / (8 * (radius (p 1) + d) ^ 2)| ≤ 1 / (4 * d) := by |
| have hr := radius_nonneg_of_mem_wedge hp |
| have hD := denominator_pos_of_mem_wedge hd hp |
| have hw := (mem_wedge_iff_walls p).mp hp |
| have helper (Y : ℝ) (hl : -radius (p 1) / 2 ≤ Y) (hu : Y ≤ radius (p 1) / 2) : |
| |(2 * Y + 2 * d + radius (p 1)) / (8 * (radius (p 1) + d) ^ 2)| ≤ 1 / (4 * d) := by |
| have hn : 0 ≤ 2 * Y + 2 * d + radius (p 1) := by linarith |
| rw [abs_of_nonneg (div_nonneg hn (by positivity))] |
| apply (div_le_div_iff₀ (by positivity : 0 < 8 * (radius (p 1) + d) ^ 2) |
| (by positivity : 0 < 4 * d)).2 |
| nlinarith [mul_nonneg (show 0 ≤ radius (p 1) - 2 * Y by linarith) hd.le, |
| mul_nonneg hr hD.le] |
| exact ⟨helper (p 0) (by linarith) (by linarith), |
| by simpa only [mul_neg, neg_mul] using helper (-p 0) (by linarith) (by linarith)⟩ |
|
|
| theorem global_tiltedMap_mapsTo {d s : ℝ} (hd : 0 < d) (hs : |s| < d) : |
| MapsTo (tiltedMap d s) wedge wedge := by |
| intro p hp |
| obtain ⟨hplus, hminus⟩ := global_wall_coefficients hd hp |
| have hsmall : |s| * (1 / (4 * d)) < 1 := by |
| rw [mul_one_div] |
| apply (div_lt_one (by positivity : 0 < 4 * d)).2 |
| linarith [abs_nonneg s] |
| have hpa : |s * ((2 * p 0 + 2 * d + radius (p 1)) / |
| (8 * (radius (p 1) + d) ^ 2))| < 1 := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_left hplus (abs_nonneg _)).trans_lt hsmall |
| have hma : |s * ((-2 * p 0 + 2 * d + radius (p 1)) / |
| (8 * (radius (p 1) + d) ^ 2))| < 1 := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_left hminus (abs_nonneg _)).trans_lt hsmall |
| rw [mem_wedge_iff_walls] at hp ⊢ |
| rw [tiltedMap_wall_plus d s (denominator_pos_of_mem_wedge hd |
| ((mem_wedge_iff_walls p).mpr hp)).ne', |
| tiltedMap_wall_minus d s (denominator_pos_of_mem_wedge hd |
| ((mem_wedge_iff_walls p).mpr hp)).ne'] |
| constructor |
| · exact mul_nonneg hp.1 (by linarith [(abs_lt.mp hpa).2]) |
| · exact mul_nonneg hp.2 (by linarith [(abs_lt.mp hma).1]) |
|
|
| theorem rational_global_inverse_of_lipschitz {d L s : ℝ} (hd : 0 < d) |
| (hL : 0 ≤ L) |
| (hlip : ∀ p ∈ wedge, ∀ q ∈ wedge, ‖field d p - field d q‖ ≤ L * ‖p - q‖) |
| (hsd : |s| < d) (hsL : |s| * L < 1) {y : Plane} (hy : y ∈ wedge) : |
| ∃! p : Plane, p ∈ wedge ∧ tiltedMap d s p = y := by |
| apply CylinderCovering.ThresholdInversion.existsUnique_inverse_on_convex |
| rational_wedge_isClosed.isComplete rational_wedge_convex |
| (mul_nonneg (abs_nonneg s) hL) hsL (g := fun p => s • field d p) ?_ ?_ hy |
| · intro p hp q hq |
| rw [← smul_sub, norm_smul, Real.norm_eq_abs] |
| calc |
| |s| * ‖field d p - field d q‖ ≤ |s| * (L * ‖p - q‖) := |
| mul_le_mul_of_nonneg_left (hlip p hp q hq) (abs_nonneg _) |
| _ = |s| * L * ‖p - q‖ := by ring |
| · intro p hp |
| have h := global_tiltedMap_mapsTo hd (s := -s) (by simpa using hsd) hp |
| simpa only [tiltedMap, neg_smul, sub_eq_add_neg] using h |
|
|
| theorem rational_global_inverse {d s : ℝ} (hd : 0 < d) (hs : |s| < d / 4) |
| {y : Plane} (hy : y ∈ wedge) : |
| ∃! p : Plane, p ∈ wedge ∧ tiltedMap d s p = y := by |
| apply rational_global_inverse_of_lipschitz hd (L := 2 / d) (by positivity) |
| · intro p hp q hq |
| have h := (field_lipschitzOn_wedge hd).dist_le_mul p hp q hq |
| change dist (field d p) (field d q) ≤ (2 / d) * dist p q at h |
| simpa only [dist_eq_norm] using h |
| · linarith |
| · rw [show |s| * (2 / d) = (2 * |s|) / d by ring] |
| apply (div_lt_one hd).2 |
| linarith |
| · exact hy |
|
|
| theorem globalWedgeInversion_main : GlobalWedgeInversionStatement := by |
| intro d hd |
| refine ⟨d / 4, by positivity, ?_⟩ |
| intro s hs |
| have hmap := global_tiltedMap_mapsTo hd (s := s) (by linarith) |
| refine ⟨hmap, ?_, ?_⟩ |
| · intro p hp q hq hpq |
| have h := rational_global_inverse hd hs (hmap hp) |
| exact h.unique ⟨hp, rfl⟩ ⟨hq, hpq.symm⟩ |
| · intro y hy |
| obtain ⟨p, hp, _⟩ := rational_global_inverse hd hs hy |
| exact ⟨p, hp.1, hp.2⟩ |
|
|
| end SlopeRational |
|
|
| end OAI |
|
|
