import Mathlib

namespace OAI

noncomputable section

open scoped Matrix.Norms.L2Operator

universe u

namespace Paper256

abbrev Mat (n : ℕ) (R : Type u) := Matrix (Fin n) (Fin n) R

abbrev Sym (n : ℕ) := selfAdjoint (Mat n ℝ)

instance instFiniteDimensionalSym (n : ℕ) : FiniteDimensional ℝ (Sym n) :=
  FiniteDimensional.finiteDimensional_submodule (selfAdjoint.submodule ℝ (Mat n ℝ))

abbrev Ambient := (Sym 4 × Sym 4) × (Fin 3 → ℝ)

abbrev SymIndex := {ij : Fin 4 × Fin 4 // ij.1 ≤ ij.2}

abbrev Coord := (SymIndex ⊕ SymIndex) ⊕ Fin 3

def qMatrix {R : Type u} [CommRing R] (y : Fin 3 → R) : Mat 3 R :=
  fun i j => if i = j then y i ^ 2 + y (i + 1) ^ 2 else -(y i * y j)

def phi {R : Type u} [CommRing R] (y : Fin 3 → R) (X : Mat 4 R) : Mat 4 R :=
  fun i j => Fin.cases
    (Fin.cases
      (Matrix.trace (qMatrix y * X.submatrix Fin.succ Fin.succ))
      (fun j => -(∑ k, X 0 k.succ * qMatrix y k j)) j)
    (fun i => Fin.cases
      (-(∑ k, qMatrix y i k * X k.succ 0))
      (fun j => X 0 0 * qMatrix y i j) j) i

def matrixValue {R : Type u} [CommRing R]
    (X Z : Mat 4 R) (y : Fin 3 → R) : R :=
  Matrix.det ((Matrix.det X) • Z - phi y (Matrix.adjugate X))

def symmetricCoordinates {R : Type u} (v : SymIndex → R) : Mat 4 R :=
  fun i j => if h : i ≤ j then v ⟨(i, j), h⟩
    else v ⟨(j, i), le_of_lt (lt_of_not_ge h)⟩

def polynomial : MvPolynomial Coord ℝ :=
  matrixValue
    (symmetricCoordinates fun ij => MvPolynomial.X (Sum.inl (Sum.inl ij)))
    (symmetricCoordinates fun ij => MvPolynomial.X (Sum.inl (Sum.inr ij)))
    (fun i => MvPolynomial.X (Sum.inr i))

def coordinates (x : Ambient) : Coord → ℝ :=
  Sum.elim (Sum.elim
    (fun ij => (x.1.1 : Mat 4 ℝ) ij.val.1 ij.val.2)
    (fun ij => (x.1.2 : Mat 4 ℝ) ij.val.1 ij.val.2)) x.2

def basePoint : Ambient := ((1, 1), 0)

def linePolynomial (x : Ambient) : Polynomial ℝ :=
  MvPolynomial.eval₂ Polynomial.C
    (fun i => Polynomial.C (coordinates basePoint i) * Polynomial.X -
      Polynomial.C (coordinates x i)) polynomial

def cone : Set Ambient :=
  {x | ∀ t : ℂ, (linePolynomial x).aeval t = 0 → t.im = 0 ∧ 0 ≤ t.re}

theorem main_result :
    Module.finrank ℝ Ambient = 23 ∧
    polynomial.IsHomogeneous 20 ∧
    polynomial.totalDegree = 20 ∧
    MvPolynomial.eval (coordinates basePoint) polynomial = 1 ∧
    (∀ (x : Ambient) (z : ℂ), (linePolynomial x).aeval z = 0 → z.im = 0) ∧
    ∀ (N : ℕ) (_hN : 0 < N) (L : Ambient →ₗ[ℝ] Sym N),
      cone ≠ {x | (L x : Mat N ℝ).PosSemidef} := by
  sorry

end Paper256

end

end OAI