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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 | import OAI.Geometry.ProjectionVolume.Model
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
namespace OAI
namespace Paper092
theorem simplex_constant_product_ratio (r s : ℕ) (hr : 0 < r) (hs : 0 < s) :
simplexConstant r * simplexConstant s / simplexConstant (r + s) =
((r + 1 : ℝ) * (s + 1) / (r + s + 1)) * (Nat.choose (r + s) r : ℝ) *
((r : ℝ) ^ r * (s : ℝ) ^ s / ((r + s : ℕ) : ℝ) ^ (r + s)) := by
have hfac := Nat.choose_mul_factorial_mul_factorial (show r ≤ r + s by omega)
simp only [Nat.add_sub_cancel_left] at hfac
have hfac' : (Nat.choose (r + s) r : ℝ) * r.factorial * s.factorial =
((r + s).factorial : ℝ) := by exact_mod_cast hfac
have hr0 : (r.factorial : ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero r
have hs0 : (s.factorial : ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero s
have hrs0 : ((r + s : ℕ) : ℝ) ≠ 0 := by positivity
have hrs1 : (r : ℝ) + s + 1 ≠ 0 := by positivity
have hfrs0 : ((r + s).factorial : ℝ) ≠ 0 := by
exact_mod_cast Nat.factorial_ne_zero (r + s)
unfold simplexConstant
push_cast
field_simp
nlinarith [hfac']
theorem dimension_twenty_ratio :
simplexConstant 10 * simplexConstant 10 / simplexConstant 20 =
(22355476 : ℝ) / 22020096 := by
norm_num [simplexConstant, Nat.factorial]
theorem dimension_twenty_strict :
simplexConstant 20 < simplexConstant 10 * simplexConstant 10 := by
norm_num [simplexConstant, Nat.factorial]
end Paper092
end OAI
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