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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import OAI.Geometry.EinsteinFour.ComplexPairAlignment
import Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality
namespace OAI
/-! The triangle inequality for the actual Fubini--Study distance on CP2. -/
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
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