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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 | import OAI.Geometry.EinsteinFour.CP2UnitGeometry
namespace OAI
/-! Unit representatives with nonnegative real overlap, including orthogonal pairs. -/
namespace PositiveEinsteinFour.Geometry
open EinsteinFour
noncomputable section
theorem cp2_aligned_unit_representative (p q : ComplexProjective2) :
∃ v : CP2Vector, ∃ hv : v ≠ 0, ‖v‖ = 1 ∧ Projectivization.mk ℂ v hv = q ∧
inner ℂ (cp2UnitRep p) v = (cp2Overlap p q : ℂ) := by
by_cases hpos : 0 < cp2Overlap p q
· exact cp2_unit_aligned p q hpos
have hz : cp2Overlap p q = 0 := le_antisymm (le_of_not_gt hpos) (cp2Overlap_nonneg p q)
refine ⟨cp2UnitRep q,cp2UnitRep_ne_zero q,cp2UnitRep_norm q,cp2UnitRep_mk q,?_⟩
have hi : inner ℂ (cp2UnitRep p) (cp2UnitRep q) = 0 := norm_eq_zero.mp hz
rw [hi,hz,Complex.ofReal_zero]
end
end PositiveEinsteinFour.Geometry
end OAI
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