| import OAI.Geometry.ProjectionBody.FacetCoordinates |
| import Mathlib.Tactic.Abel |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped RealInnerProductSpace |
|
|
| namespace ProjectionCounterexample |
|
|
| / |
| def graphLinear {n : ℕ} (i : Fin (n + 1)) (a : E n) : E n →ₗ[ℝ] E (n + 1) where |
| toFun x := insertZero i x - ⟪a, x⟫ • EuclideanSpace.single i 1 |
| map_add' x y := by |
| simp only [map_add, inner_add_right, add_smul] |
| abel |
| map_smul' c x := by |
| simp only [map_smul, real_inner_smul_right, smul_sub, smul_smul, RingHom.id_apply] |
|
|
| / |
| def graphNormal {n : ℕ} (i : Fin (n + 1)) (a : E n) : E (n + 1) := |
| insertZero i a + EuclideanSpace.single i 1 |
|
|
| @[simp] theorem graphNormal_same {n : ℕ} (i : Fin (n + 1)) (a : E n) : |
| graphNormal i a i = 1 := by simp [graphNormal] |
|
|
| theorem graphNormal_ne_zero {n : ℕ} (i : Fin (n + 1)) (a : E n) : |
| graphNormal i a ≠ 0 := by |
| intro h |
| have := congrArg (fun x : E (n + 1) => x i) h |
| simp at this |
|
|
| @[simp] theorem graphLinear_same {n : ℕ} (i : Fin (n + 1)) (a x : E n) : |
| graphLinear i a x i = -⟪a, x⟫ := by simp [graphLinear] |
|
|
| @[simp] theorem graphLinear_succAbove {n : ℕ} (i : Fin (n + 1)) |
| (a x : E n) (j : Fin n) : |
| graphLinear i a x (i.succAbove j) = x j := by simp [graphLinear] |
|
|
| theorem graphLinear_inner {n : ℕ} (i : Fin (n + 1)) (a x y : E n) : |
| ⟪graphLinear i a x, graphLinear i a y⟫ = ⟪x, y⟫ + ⟪a, x⟫ * ⟪a, y⟫ := by |
| simp [graphLinear, inner_sub_left, inner_sub_right, real_inner_smul_left, |
| real_inner_smul_right, EuclideanSpace.inner_single_left, |
| EuclideanSpace.inner_single_right, (insertZero i).inner_map_map, mul_comm] |
|
|
| theorem graphNormal_inner_graphLinear {n : ℕ} (i : Fin (n + 1)) (a x : E n) : |
| ⟪graphNormal i a, graphLinear i a x⟫ = 0 := by |
| simp [graphNormal, graphLinear, inner_add_left, inner_sub_right, |
| real_inner_smul_right, EuclideanSpace.inner_single_left, |
| EuclideanSpace.inner_single_right, (insertZero i).inner_map_map] |
|
|
| theorem graphNormal_norm_sq {n : ℕ} (i : Fin (n + 1)) (a : E n) : |
| ‖graphNormal i a‖ ^ 2 = 1 + ‖a‖ ^ 2 := by |
| calc |
| ‖graphNormal i a‖ ^ 2 = ⟪graphNormal i a, graphNormal i a⟫ := |
| (real_inner_self_eq_norm_sq _).symm |
| _ = 1 + ‖a‖ ^ 2 := by |
| unfold graphNormal |
| rw [inner_add_left, inner_add_right, inner_add_right] |
| simp [EuclideanSpace.inner_single_left, EuclideanSpace.inner_single_right, |
| add_comm] |
|
|
| end ProjectionCounterexample |
|
|
| end |
|
|
| end OAI |
|
|
