File size: 4,284 Bytes
cead56f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | 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
|