import OAI.Computability.UniqueGames.Quadratic.AlignmentCertificateLemmas
import OAI.Computability.UniqueGames.Quadratic.Block

namespace OAI

section

/-!
# The binary trace annihilator of a quadratic-block hyperplane

The trace annihilator of the field hyperplane perpendicular to a nonzero
vector is exactly its field line.  Scaling test vectors first upgrades binary
trace orthogonality to field-valued orthogonality; explicit coordinate test
vectors then identify the annihilator.  No earlier gadget result is used.
-/

namespace UniqueGamesTheorem.Quadratic

variable {F : Type*} [Field F]

private theorem dot_sub_right (v y z : Vec F) :
    dot v (y - z) = dot v y - dot v z := by
  simp [dot]
  ring

private theorem dot_single (v : Vec F) (i : Fin 3) (a : F) :
    dot v (Pi.single i a) = v i * a := by
  fin_cases i <;> simp [dot]

/-- The field-valued annihilator of a nonzero vector's perpendicular
hyperplane is the line generated by that vector. -/
theorem annihilator_hyperplane_iff_mem_line {v z : Vec F} (hv : v ≠ 0) :
    (∀ y ∈ hyperplane v, dot z y = 0) ↔ z ∈ line v := by
  constructor
  · intro hz
    have hv' : ∃ i : Fin 3, v i ≠ 0 := by
      by_contra! h
      exact hv (funext h)
    obtain ⟨i, hi⟩ := hv'
    apply (mem_line_iff v z).mpr
    refine ⟨z i / v i, ?_⟩
    ext j
    have hy : Pi.single j (1 : F) - (v j / v i) • Pi.single i (1 : F) ∈
        hyperplane v := by
      rw [mem_hyperplane, dot_sub_right, dot_smul_right, dot_single, dot_single]
      simp [div_mul_cancel₀ _ hi]
    have hzero := hz _ hy
    rw [dot_sub_right, dot_smul_right, dot_single, dot_single,
      mul_one, mul_one, sub_eq_zero] at hzero
    change (z i / v i) * v j = z j
    rw [hzero]
    simp only [div_eq_mul_inv]
    ring
  · intro hz y hy
    obtain ⟨t, rfl⟩ := (mem_line_iff v z).mp hz
    rw [dot_smul_left, (mem_hyperplane v y).mp hy, mul_zero]

variable [Finite F] [Algebra (ZMod 2) F]

/-- Exact trace-annihilator identity for the field hyperplane.  Trace
nondegeneracy is provided by the proved finite-field trace pairing theorem. -/
theorem trace_annihilator_hyperplane_iff_mem_line {v z : Vec F} (hv : v ≠ 0) :
    (∀ y ∈ hyperplane v, traceBinary (dot z y) = 0) ↔ z ∈ line v := by
  rw [← annihilator_hyperplane_iff_mem_line hv]
  constructor
  · intro hz y hy
    apply (trace_mul_vanish_iff (dot z y)).mp
    intro t
    have ht := hz (t • y) ((hyperplane v).smul_mem t hy)
    simpa only [dot_smul_right] using ht
  · intro hz y hy
    rw [hz y hy, map_zero]

/-- The same annihilator identity in set notation, using the actual trace. -/
theorem trace_annihilator_hyperplane (v : Vec F) (hv : v ≠ 0) :
    {z : Vec F | ∀ y ∈ hyperplane v, Algebra.trace (ZMod 2) F (dot z y) = 0} =
      (line v : Set (Vec F)) := by
  ext z
  exact trace_annihilator_hyperplane_iff_mem_line hv

end UniqueGamesTheorem.Quadratic

end

end OAI