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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 | import OAI.Geometry.ProjectionVolume.ProductFacetAreas
import OAI.Geometry.ProjectionVolume.ProductFacetGeometry
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped ENNReal
namespace Paper092
theorem splitEuclideanProduct_ten_ten : splitEuclideanProduct 10 10 = splitBlocks := rfl
theorem productFacet_inl_some_area (i : Fin 10) :
μHE[19] (productFacet (.inl (some i))) =
ENNReal.ofReal (1 / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ))) := by
rw [productFacet_inl_eq_preimage, ← splitEuclideanProduct_ten_ten]
rw [simplexProduct_coordinate_facet_area_left 9 10 i]
congr 1
simp only [one_div, mul_inv_rev]
ring
theorem productFacet_inr_some_area (i : Fin 10) :
μHE[19] (productFacet (.inr (some i))) =
ENNReal.ofReal (1 / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ))) := by
rw [productFacet_inr_eq_preimage, ← splitEuclideanProduct_ten_ten]
rw [simplexProduct_coordinate_facet_area_right 10 9 i]
congr 1
simp only [one_div, mul_inv_rev]
theorem productFacet_inl_none_area :
μHE[19] (productFacet (.inl none)) =
ENNReal.ofReal (Real.sqrt 10 / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ))) := by
rw [productFacet_inl_eq_preimage, ← splitEuclideanProduct_ten_ten]
rw [simplexProduct_diagonal_facet_area_left 9 10]
congr 1
norm_num only [Nat.cast_ofNat]
simp only [div_eq_mul_inv]
ring
theorem productFacet_inr_none_area :
μHE[19] (productFacet (.inr none)) =
ENNReal.ofReal (Real.sqrt 10 / ((Nat.factorial 9 : ℝ) * (Nat.factorial 10 : ℝ))) := by
rw [productFacet_inr_eq_preimage, ← splitEuclideanProduct_ten_ten]
rw [simplexProduct_diagonal_facet_area_right 10 9]
congr 1
norm_num only [Nat.cast_ofNat]
simp only [div_eq_mul_inv]
ring
end Paper092
end
end OAI
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