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9eeb9c7 | 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 | import OAI.Analysis.HyperbolicCones.FormRigidity
namespace OAI
/-! The sign-vector coefficient identity for the Choi–Lam form. -/
noncomputable section
namespace Paper256
theorem form_sign_expansion
(l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ)
(σ : Fin 3 → ℝ) (hσ : ∀ i, σ i = 1 ∨ σ i = -1) :
l σ σ = (∑ i, l (Pi.single i 1) (Pi.single i 1)) +
(l (Pi.single 0 1) (Pi.single 1 1) + l (Pi.single 1 1) (Pi.single 0 1)) * σ 0 * σ 1 +
(l (Pi.single 0 1) (Pi.single 2 1) + l (Pi.single 2 1) (Pi.single 0 1)) * σ 0 * σ 2 +
(l (Pi.single 1 1) (Pi.single 2 1) + l (Pi.single 2 1) (Pi.single 1 1)) * σ 1 * σ 2 := by
have hs (i : Fin 3) : σ i ^ 2 = 1 := by
rcases hσ i with hi | hi <;> simp [hi]
rw [form_bilinear_coordinates]
simp [Fin.sum_univ_succ]
linear_combination
l (Pi.single 0 1) (Pi.single 0 1) * hs 0 +
l (Pi.single 1 1) (Pi.single 1 1) * hs 1 +
l (Pi.single 2 1) (Pi.single 2 1) * hs 2
theorem dominated_form_sign_relation
(l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ)
(h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y)
(σ : Fin 3 → ℝ) (hσ : ∀ i, σ i = 1 ∨ σ i = -1) :
0 = (∑ i, l (Pi.single i 1) (Pi.single i 1)) +
(l (Pi.single 0 1) (Pi.single 1 1) + l (Pi.single 1 1) (Pi.single 0 1)) * σ 0 * σ 1 +
(l (Pi.single 0 1) (Pi.single 2 1) + l (Pi.single 2 1) (Pi.single 0 1)) * σ 0 * σ 2 +
(l (Pi.single 1 1) (Pi.single 2 1) + l (Pi.single 2 1) (Pi.single 1 1)) * σ 1 * σ 2 := by
rw [← form_sign_expansion l σ hσ]
exact (dominated_form_sign_zero l h σ hσ).symm
end Paper256
end
end OAI
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