| import OAI.Geometry.ProjectionVolume.ProductNormals |
| import OAI.Geometry.ProjectionVolume.ProductCoefficients |
| import OAI.Geometry.ProjectionVolume.ProductFacetAreaValues |
| import OAI.Geometry.ProjectionVolume.ProjectionJacobian |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| namespace Paper092 |
|
|
| theorem productFacet_brightness (a : ProductFacetIndex) (u : Euclidean 20) : |
| brightness (productFacet a) u = |
| |productVelocity u a| / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ)) := by |
| have hsqrt : 0 < Real.sqrt (10 : ℝ) := Real.sqrt_pos.mpr (by norm_num) |
| cases a with |
| | inl a => |
| cases a with |
| | none => |
| have hn : ‖liftFirst (diagonalUnitNormal 10)‖ = 1 := by |
| rw [liftFirst_norm] |
| exact diagonalUnitNormal_norm 10 (by norm_num) |
| rw [brightness_of_subset_affine_hyperplane u (liftFirst (diagonalUnitNormal 10)) |
| hn (Real.sqrt 10)⁻¹ _ productFacet_left_diagonal_plane] |
| rw [show 20 - 1 = 19 by rfl, productFacet_inl_none_area, |
| ENNReal.toReal_ofReal (by positivity), inner_liftFirst, inner_diagonalUnitNormal] |
| simp only [productVelocity, simplexVelocity, abs_neg, abs_div, Nat.cast_ofNat, |
| abs_of_pos hsqrt] |
| field_simp |
| | some i => |
| have hn : ‖liftFirst (EuclideanSpace.single i (1 : ℝ))‖ = 1 := by |
| rw [liftFirst_norm, coordinate_normal_norm] |
| rw [brightness_of_subset_affine_hyperplane u (liftFirst (EuclideanSpace.single i 1)) |
| hn 0 _ (productFacet_left_coordinate_plane i)] |
| rw [show 20 - 1 = 19 by rfl, productFacet_inl_some_area, |
| ENNReal.toReal_ofReal (by positivity)] |
| simp [inner_liftFirst, EuclideanSpace.inner_single_right, productVelocity, |
| simplexVelocity, div_eq_mul_inv] |
| | inr a => |
| cases a with |
| | none => |
| have hn : ‖liftSecond (diagonalUnitNormal 10)‖ = 1 := by |
| rw [liftSecond_norm] |
| exact diagonalUnitNormal_norm 10 (by norm_num) |
| rw [brightness_of_subset_affine_hyperplane u (liftSecond (diagonalUnitNormal 10)) |
| hn (Real.sqrt 10)⁻¹ _ productFacet_right_diagonal_plane] |
| rw [show 20 - 1 = 19 by rfl, productFacet_inr_none_area, |
| ENNReal.toReal_ofReal (by positivity), inner_liftSecond, inner_diagonalUnitNormal] |
| simp only [productVelocity, simplexVelocity, abs_neg, abs_div, Nat.cast_ofNat, |
| abs_of_pos hsqrt] |
| field_simp |
| | some i => |
| have hn : ‖liftSecond (EuclideanSpace.single i (1 : ℝ))‖ = 1 := by |
| rw [liftSecond_norm, coordinate_normal_norm] |
| rw [brightness_of_subset_affine_hyperplane u (liftSecond (EuclideanSpace.single i 1)) |
| hn 0 _ (productFacet_right_coordinate_plane i)] |
| rw [show 20 - 1 = 19 by rfl, productFacet_inr_some_area, |
| ENNReal.toReal_ofReal (by positivity)] |
| simp [inner_liftSecond, EuclideanSpace.inner_single_right, productVelocity, |
| simplexVelocity, div_eq_mul_inv] |
|
|
| theorem productWitness_brightness (u : Euclidean 20) : |
| brightness productWitness u = |
| ((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i| + |
| (∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|) / |
| (2 * (Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ)) := by |
| by_cases hu : u = 0 |
| · simp [hu, brightness, firstBlock, secondBlock] |
| calc |
| brightness productWitness u = |
| ∑ a : {a : ProductFacetIndex // productVelocity u a < 0}, |
| brightness (productFacet a.val) u := by |
| rw [brightness_product_eq_front_sum u hu, Finset.mul_sum] |
| rfl |
| _ = ∑ a : {a : ProductFacetIndex // productVelocity u a < 0}, |
| -productVelocity u a.val / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ)) := by |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [productFacet_brightness, abs_of_neg a.property] |
| _ = (∑ a : {a : ProductFacetIndex // productVelocity u a < 0}, |
| -productVelocity u a.val) / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ)) := by |
| rw [Finset.sum_div] |
| _ = ((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i| + |
| (∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|) / |
| (2 * (Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ)) := by |
| rw [product_front_coefficient] |
| ring |
|
|
| end Paper092 |
|
|
| end |
|
|
| end OAI |
|
|
