| import OAI.Analysis.HyperbolicCones.FormBasic | |
| namespace OAI | |
| /-! A bilinear square dominated by the Choi--Lam form vanishes. -/ | |
| noncomputable section | |
| open scoped Matrix.Norms.L2Operator | |
| universe u | |
| namespace Paper256 | |
| theorem form_linear_coordinates {W : Type u} [AddCommGroup W] [Module ℝ W] | |
| (f : (Fin 3 → ℝ) →ₗ[ℝ] W) (x : Fin 3 → ℝ) : | |
| f x = ∑ i, x i • f (Pi.single i 1) := by | |
| have hx : x = ∑ i, x i • (Pi.single i 1 : Fin 3 → ℝ) := by | |
| ext j | |
| simp [Finset.sum_apply, Pi.single_apply] | |
| conv_lhs => rw [hx] | |
| simp | |
| theorem form_bilinear_coordinates | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) (z y : Fin 3 → ℝ) : | |
| l z y = ∑ i, ∑ j, (z i * y j) * l (Pi.single i 1) (Pi.single j 1) := by | |
| rw [form_linear_coordinates l z] | |
| simp only [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [form_linear_coordinates (l (Pi.single i 1)) y] | |
| simp [Finset.mul_sum, mul_assoc] | |
| theorem dominated_form_vanishes_at_zero | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) | |
| (z y : Fin 3 → ℝ) (hz : choiLam z y = 0) : l z y = 0 := by | |
| have hh := h z y | |
| rw [hz] at hh | |
| nlinarith [sq_nonneg (l z y)] | |
| theorem dominated_form_cyclic_coefficient | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (i : Fin 3) : | |
| l (Pi.single i 1) (Pi.single (i + 2) 1) = 0 := | |
| dominated_form_vanishes_at_zero l h _ _ (choiLam_cyclic_zero i) | |
| theorem dominated_form_sign_zero | |
| (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) : l σ σ = 0 := | |
| dominated_form_vanishes_at_zero l h σ σ (choiLam_sign_zero σ hσ) | |
| theorem dominated_form_symmetric_coefficients | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) : | |
| (l (Pi.single 0 1) (Pi.single 1 1) + l (Pi.single 1 1) (Pi.single 0 1) = 0) ∧ | |
| (l (Pi.single 0 1) (Pi.single 2 1) + l (Pi.single 2 1) (Pi.single 0 1) = 0) ∧ | |
| (l (Pi.single 1 1) (Pi.single 2 1) + l (Pi.single 2 1) (Pi.single 1 1) = 0) := by | |
| have h0 := dominated_form_vanishes_at_zero l h ![1, 1, 1] ![1, 1, 1] | |
| (by simp [choiLam_expand, Matrix.cons_val]; norm_num) | |
| have h1 := dominated_form_vanishes_at_zero l h ![1, 1, -1] ![1, 1, -1] | |
| (by simp [choiLam_expand, Matrix.cons_val]; norm_num) | |
| have h2 := dominated_form_vanishes_at_zero l h ![1, -1, 1] ![1, -1, 1] | |
| (by simp [choiLam_expand, Matrix.cons_val]; norm_num) | |
| have h3 := dominated_form_vanishes_at_zero l h ![-1, 1, 1] ![-1, 1, 1] | |
| (by simp [choiLam_expand, Matrix.cons_val]; norm_num) | |
| rw [form_bilinear_coordinates] at h0 h1 h2 h3 | |
| simp [Fin.sum_univ_succ] at h0 h1 h2 h3 | |
| constructor | |
| · linarith | |
| constructor <;> linarith | |
| theorem dominated_form_off_diagonal | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (i j : Fin 3) (hij : i ≠ j) : | |
| l (Pi.single i 1) (Pi.single j 1) = 0 := by | |
| obtain ⟨h01, h02, h12⟩ := dominated_form_symmetric_coefficients l h | |
| have h20 := dominated_form_cyclic_coefficient l h 0 | |
| have h10 := dominated_form_cyclic_coefficient l h 1 | |
| have h21 := dominated_form_cyclic_coefficient l h 2 | |
| simp only [Fin.reduceAdd] at h20 h10 h21 | |
| fin_cases i <;> fin_cases j <;> simp_all [Fin.reduceEq] | |
| theorem form_curve_of_off_diagonal | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (hoff : ∀ i j : Fin 3, i ≠ j → l (Pi.single i 1) (Pi.single j 1) = 0) | |
| (i : Fin 3) (s : ℝ) : | |
| l ((Pi.single i 1 : Fin 3 → ℝ) + s • Pi.single (i + 2) 1) | |
| (Pi.single (i + 2) 1 + s • Pi.single i 1) = | |
| s * (l (Pi.single i 1) (Pi.single i 1) + | |
| l (Pi.single (i + 2) 1) (Pi.single (i + 2) 1)) := by | |
| have hi : i ≠ i + 2 := by fin_cases i <;> decide | |
| simp [map_add, map_smul, LinearMap.add_apply, LinearMap.smul_apply, | |
| hoff i (i + 2) hi, hoff (i + 2) i hi.symm] | |
| ring | |
| theorem dominated_form_diagonal_formula | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (z y : Fin 3 → ℝ) : | |
| l z y = ∑ i, z i * y i * l (Pi.single i 1) (Pi.single i 1) := by | |
| rw [form_bilinear_coordinates] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [Finset.sum_eq_single i] | |
| · intro j _ hji | |
| simp [dominated_form_off_diagonal l h i j (Ne.symm hji)] | |
| · simp | |
| theorem dominated_form_diagonal_pair_bound | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (i : Fin 3) (s : ℝ) (hs : s ≠ 0) : | |
| (l (Pi.single i 1) (Pi.single i 1) + | |
| l (Pi.single (i + 2) 1) (Pi.single (i + 2) 1)) ^ 2 ≤ s ^ 2 := by | |
| have hh := h ((Pi.single i 1 : Fin 3 → ℝ) + s • Pi.single (i + 2) 1) | |
| (Pi.single (i + 2) 1 + s • Pi.single i 1) | |
| rw [choiLam_curve, form_curve_of_off_diagonal l (dominated_form_off_diagonal l h)] at hh | |
| apply le_of_mul_le_mul_left (a := s ^ 2) _ (sq_pos_of_ne_zero hs) | |
| nlinarith only [hh] | |
| theorem dominated_form_diagonal_pair_zero | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (i : Fin 3) : | |
| l (Pi.single i 1) (Pi.single i 1) + | |
| l (Pi.single (i + 2) 1) (Pi.single (i + 2) 1) = 0 := by | |
| let c := l (Pi.single i 1) (Pi.single i 1) + | |
| l (Pi.single (i + 2) 1) (Pi.single (i + 2) 1) | |
| change c = 0 | |
| by_contra hc | |
| have hh := dominated_form_diagonal_pair_bound l h i (c / 2) (div_ne_zero hc (by norm_num)) | |
| change c ^ 2 ≤ (c / 2) ^ 2 at hh | |
| nlinarith [sq_pos_of_ne_zero hc] | |
| theorem dominated_form_diagonal_zero | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (i : Fin 3) : | |
| l (Pi.single i 1) (Pi.single i 1) = 0 := by | |
| have h0 := dominated_form_diagonal_pair_zero l h 0 | |
| have h1 := dominated_form_diagonal_pair_zero l h 1 | |
| have h2 := dominated_form_diagonal_pair_zero l h 2 | |
| simp only [Fin.reduceAdd] at h0 h1 h2 | |
| fin_cases i <;> simp only [Fin.reduceFinMk] <;> linarith | |
| theorem dominated_form_coefficient_zero | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) (i j : Fin 3) : | |
| l (Pi.single i 1) (Pi.single j 1) = 0 := by | |
| by_cases hij : i = j | |
| · subst j | |
| exact dominated_form_diagonal_zero l h i | |
| · exact dominated_form_off_diagonal l h i j hij | |
| theorem choiLam_no_dominated_square | |
| (l : (Fin 3 → ℝ) →ₗ[ℝ] (Fin 3 → ℝ) →ₗ[ℝ] ℝ) | |
| (h : ∀ z y, (l z y) ^ 2 ≤ choiLam z y) : l = 0 := by | |
| apply LinearMap.ext | |
| intro z | |
| apply LinearMap.ext | |
| intro y | |
| change l z y = 0 | |
| rw [form_bilinear_coordinates] | |
| simp [dominated_form_coefficient_zero l h] | |
| end Paper256 | |
| end | |
| end OAI | |