| import OAI.Geometry.HeilbronnTriangle.PrimitiveNormal |
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| namespace OAI |
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| namespace Problem355.CrossNormal |
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| open Matrix |
| open scoped Matrix |
| open PrimitiveNormal |
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| lemma toEuclidean_zsmul (g : ℤ) (w : Vector) : |
| toEuclidean (g • w) = (g : ℝ) • toEuclidean w := by |
| ext i |
| simp [PrimitiveNormal.toEuclidean, Pi.smul_apply, smul_eq_mul] |
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| theorem norm_cross_le (u v : Vector) : |
| ‖toEuclidean (u ⨯₃ v)‖ ≤ ‖toEuclidean u‖ * ‖toEuclidean v‖ := by |
| let d : ℝ := ∑ i : Fin 3, (u i : ℝ) * (v i : ℝ) |
| have hlagrange : ‖toEuclidean (u ⨯₃ v)‖ ^ 2 + d ^ 2 = |
| ‖toEuclidean u‖ ^ 2 * ‖toEuclidean v‖ ^ 2 := by |
| simp only [EuclideanSpace.real_norm_sq_eq] |
| simp [PrimitiveNormal.toEuclidean, d, Fin.sum_univ_three, cross_apply] |
| ring |
| apply (sq_le_sq₀ (norm_nonneg _) |
| (mul_nonneg (norm_nonneg _) (norm_nonneg _))).mp |
| calc |
| ‖toEuclidean (u ⨯₃ v)‖ ^ 2 ≤ |
| ‖toEuclidean u‖ ^ 2 * ‖toEuclidean v‖ ^ 2 := by nlinarith [sq_nonneg d] |
| _ = (‖toEuclidean u‖ * ‖toEuclidean v‖) ^ 2 := by ring |
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| theorem norm_le_of_positive_factor {v w : Vector} {g : ℤ} |
| (hg : 0 < g) (hscale : v = g • w) : ‖toEuclidean w‖ ≤ ‖toEuclidean v‖ := by |
| have hgR : (0 : ℝ) < g := by exact_mod_cast hg |
| have hg1 : (1 : ℝ) ≤ g := by exact_mod_cast hg |
| rw [hscale, toEuclidean_zsmul, norm_smul, Real.norm_eq_abs, abs_of_pos hgR] |
| nlinarith [norm_nonneg (toEuclidean w)] |
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| theorem determinant_eq_factor_dot {u v w : Vector} {g : ℤ} |
| (hscale : u ⨯₃ v = g • w) (x : Vector) : |
| Matrix.det ![x, u, v] = g * (x ⬝ᵥ w) := by |
| rw [← triple_product_eq_det, hscale, dotProduct_smul, smul_eq_mul] |
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| theorem determinant_eq_iff_difference_dot_zero {u v w : Vector} {g : ℤ} |
| (hg : g ≠ 0) (hscale : u ⨯₃ v = g • w) (x z : Vector) : |
| Matrix.det ![x, u, v] = Matrix.det ![z, u, v] ↔ (x - z) ⬝ᵥ w = 0 := by |
| rw [determinant_eq_factor_dot hscale, determinant_eq_factor_dot hscale, |
| sub_dotProduct, sub_eq_zero] |
| constructor |
| · exact mul_left_cancel₀ hg |
| · intro h |
| rw [h] |
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| theorem exists_bounded_primitive_cross_factor (u v : Vector) (huv : u ⨯₃ v ≠ 0) : |
| ∃ g : ℤ, 0 < g ∧ ∃ w : Vector, IsPrimitive w ∧ u ⨯₃ v = g • w ∧ |
| u ⬝ᵥ w = 0 ∧ v ⬝ᵥ w = 0 ∧ |
| ‖toEuclidean w‖ ≤ ‖toEuclidean u‖ * ‖toEuclidean v‖ ∧ |
| (∀ x z : Vector, Matrix.det ![x, u, v] = Matrix.det ![z, u, v] ↔ |
| (x - z) ⬝ᵥ w = 0) := by |
| obtain ⟨g, hg, w, hw, hscale⟩ := exists_primitive_factor (u ⨯₃ v) huv |
| have horth (x : Vector) (hx : x ⬝ᵥ u ⨯₃ v = 0) : x ⬝ᵥ w = 0 := by |
| rw [hscale, dotProduct_smul, smul_eq_mul] at hx |
| exact (mul_eq_zero.mp hx).resolve_left (ne_of_gt hg) |
| refine ⟨g, hg, w, hw, hscale, horth u (dot_self_cross u v), |
| horth v (dot_cross_self u v), ?_, ?_⟩ |
| · exact (norm_le_of_positive_factor hg hscale).trans (norm_cross_le u v) |
| · exact determinant_eq_iff_difference_dot_zero (ne_of_gt hg) hscale |
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| end Problem355.CrossNormal |
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| end OAI |
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