| import OAI.Geometry.EinsteinFour.ComplexPairAlignment |
| import Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality |
|
|
| namespace OAI |
|
|
| /-! The triangle inequality for the actual Fubini |
| namespace PositiveEinsteinFour.Geometry |
| open EinsteinFour |
| noncomputable section |
|
|
| theorem complex_unit_angle (u v : CP2Vector) (hu : ‖u‖ = 1) (hv : ‖v‖ = 1) : |
| InnerProductGeometry.angle u v = Real.arccos (inner ℝ u v) := by |
| simp only [InnerProductGeometry.angle,hu,hv,one_mul,div_one] |
|
|
| theorem fubiniStudyDistance_triangle (P Q R : ComplexProjective2) : |
| fubiniStudyDistance P R ≤ fubiniStudyDistance P Q + fubiniStudyDistance Q R := by |
| obtain ⟨p,hp,hpn,hpP,hqp⟩ := cp2_aligned_unit_representative Q P |
| obtain ⟨r,hr,hrn,hrR,hqr⟩ := cp2_aligned_unit_representative Q R |
| let q := cp2UnitRep Q |
| have hpqR : inner ℝ q p = cp2Overlap Q P := by |
| simpa only [Complex.ofReal_re,← real_inner_eq_re_inner ℂ] using! congrArg Complex.re hqp |
| have hqrR : inner ℝ q r = cp2Overlap Q R := by |
| simpa only [Complex.ofReal_re,← real_inner_eq_re_inner ℂ] using! congrArg Complex.re hqr |
| have hreal : inner ℝ p r ≤ ‖inner ℂ p r‖ := by |
| simpa only [← real_inner_eq_re_inner ℂ] using! Complex.re_le_norm (inner ℂ p r) |
| calc |
| fubiniStudyDistance P R = Real.arccos ‖inner ℂ p r‖ := by |
| rw [fubiniStudyDistance_eq_arccos,cp2Overlap_of_unit_representatives P R p r hp hr hpn hrn hpP hrR] |
| _ ≤ InnerProductGeometry.angle p r := by |
| rw [complex_unit_angle p r hpn hrn] |
| exact Real.antitone_arccos hreal |
| _ ≤ InnerProductGeometry.angle p q + InnerProductGeometry.angle q r := |
| InnerProductGeometry.angle_le_angle_add_angle p q r |
| _ = fubiniStudyDistance P Q + fubiniStudyDistance Q R := by |
| rw [complex_unit_angle p q hpn (cp2UnitRep_norm Q), |
| complex_unit_angle q r (cp2UnitRep_norm Q) hrn, |
| real_inner_comm q p,hpqR,hqrR,cp2Overlap_comm Q P, |
| fubiniStudyDistance_eq_arccos,fubiniStudyDistance_eq_arccos] |
|
|
| end |
| end PositiveEinsteinFour.Geometry |
|
|
| end OAI |
|
|
