| 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 | |