| import OAI.Geometry.ProjectionBody.ConvexGeometry |
| import Mathlib.Algebra.BigOperators.Fin |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
|
|
| namespace ProjectionCounterexample |
|
|
| /-- The coordinate isometry inserting a zero in slot `i`. -/ |
| def insertZero {n : ℕ} (i : Fin (n + 1)) : E n →ₗᵢ[ℝ] E (n + 1) where |
| toFun x := WithLp.toLp 2 (i.insertNth 0 x) |
| map_add' x y := by |
| ext j |
| induction j using i.succAboveCases <;> simp |
| map_smul' c x := by |
| ext j |
| induction j using i.succAboveCases <;> simp |
| norm_map' x := by |
| apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp |
| rw [EuclideanSpace.real_norm_sq_eq, EuclideanSpace.real_norm_sq_eq, |
| Fin.sum_univ_succAbove _ i] |
| simp |
|
|
| @[simp] theorem insertZero_same {n : ℕ} (i : Fin (n + 1)) (x : E n) : |
| insertZero i x i = 0 := by simp [insertZero] |
|
|
| @[simp] theorem insertZero_succAbove {n : ℕ} (i : Fin (n + 1)) (x : E n) |
| (j : Fin n) : insertZero i x (i.succAbove j) = x j := by simp [insertZero] |
|
|
| theorem sum_insertZero {n : ℕ} (i : Fin (n + 1)) (x : E n) : |
| ∑ j, insertZero i x j = ∑ j, x j := by |
| rw [Fin.sum_univ_succAbove _ i] |
| simp |
|
|
| /-- A coordinate facet is an actual isometric image of the lower-dimensional simplex. -/ |
| theorem coordinate_facet_eq_image {n : ℕ} (i : Fin (n + 1)) : |
| {x ∈ simplex (n + 1) | x i = 0} = insertZero i '' simplex n := by |
| simp_rw [simplex_eq_coordinates] |
| apply Subset.antisymm |
| · rintro x ⟨⟨hx, hs⟩, hxi⟩ |
| let y : E n := WithLp.toLp 2 (fun j => x (i.succAbove j)) |
| have hy : (∀ j, 0 ≤ y j) ∧ ∑ j, y j ≤ 1 := by |
| refine ⟨fun j => hx _, ?_⟩ |
| rw [Fin.sum_univ_succAbove _ i, hxi, zero_add] at hs |
| exact hs |
| refine ⟨y, hy, ?_⟩ |
| ext j |
| induction j using i.succAboveCases |
| · simpa using hxi.symm |
| · simp [y] |
| · rintro x ⟨y, ⟨hy, hs⟩, rfl⟩ |
| refine ⟨⟨?_, ?_⟩, by simp⟩ |
| · intro j |
| induction j using i.succAboveCases |
| · simp |
| · simpa using hy _ |
| · simpa [sum_insertZero] using hs |
|
|
| /-- The affine chart on the facet whose coordinate sum is one. -/ |
| def sumFacetMap (n : ℕ) (x : E n) : E (n + 1) := |
| WithLp.toLp 2 (Fin.snoc (fun j => x j) (1 - ∑ j, x j)) |
|
|
| @[simp] theorem sumFacetMap_castSucc (n : ℕ) (x : E n) (i : Fin n) : |
| sumFacetMap n x i.castSucc = x i := by simp [sumFacetMap] |
|
|
| @[simp] theorem sumFacetMap_last (n : ℕ) (x : E n) : |
| sumFacetMap n x (Fin.last n) = 1 - ∑ j, x j := by simp [sumFacetMap] |
|
|
| @[simp] theorem sum_sumFacetMap (n : ℕ) (x : E n) : |
| ∑ j, sumFacetMap n x j = 1 := by |
| rw [Fin.sum_univ_castSucc] |
| simp |
|
|
| /-- The non-coordinate facet is an actual affine image of the lower-dimensional simplex. -/ |
| theorem sum_facet_eq_image (n : ℕ) : |
| {x ∈ simplex (n + 1) | ∑ j, x j = 1} = sumFacetMap n '' simplex n := by |
| simp_rw [simplex_eq_coordinates] |
| apply Subset.antisymm |
| · rintro x ⟨⟨hx, _⟩, hs⟩ |
| let y : E n := WithLp.toLp 2 (fun j : Fin n => x j.castSucc) |
| have hsum : ∑ j, y j = 1 - x (Fin.last n) := by |
| rw [Fin.sum_univ_castSucc] at hs |
| dsimp [y] |
| linarith |
| have hy : (∀ j, 0 ≤ y j) ∧ ∑ j, y j ≤ 1 := by |
| refine ⟨fun j => hx _, ?_⟩ |
| rw [hsum] |
| linarith [hx (Fin.last n)] |
| refine ⟨y, hy, ?_⟩ |
| ext j |
| induction j using Fin.lastCases |
| · simp [hsum] |
| · simp [y] |
| · rintro x ⟨y, ⟨hy, hs⟩, rfl⟩ |
| refine ⟨⟨?_, by simp⟩, by simp⟩ |
| intro j |
| induction j using Fin.lastCases |
| · simp only [sumFacetMap_last] |
| exact sub_nonneg.mpr hs |
| · simpa using hy _ |
|
|
| end ProjectionCounterexample |
|
|
| end |
|
|
| end OAI |
|
|
