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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.Trajectories.FallingMoment
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.Sparse
open Polynomial
lemma lineMoment_stein (m h : ℕ) (P : ℝ[X]) (hp : h+P.natDegree < m) :
lineMoment m h (stein m h P) = 0 := by
let L := (lineMoment m h).comp (stein m h)
change L P = 0
conv_lhs => rw [← P.sum_monomial_eq]
simp only [Polynomial.sum_def,map_sum]
apply Finset.sum_eq_zero
intro n hn
exact lineMoment_stein_monomial (show h+n < m from lt_of_le_of_lt
(Nat.add_le_add_left (le_natDegree_of_ne_zero (mem_support_iff.mp hn)) h) hp) _
lemma lineMoment_C_mul (m h : ℕ) (a : ℝ) (P : ℝ[X]) :
lineMoment m h (C a * P) = a * lineMoment m h P := by
rw [← smul_eq_C_mul,map_smul,smul_eq_mul]
def centeredLineMoment (m h u v : ℕ) : ℝ :=
lineMoment m h (X^v*(X-1)^u)
@[simp] lemma centeredLineMoment_zero (m h v : ℕ) :
centeredLineMoment m h 0 v = fallingMoment m h v := by
simp only [centeredLineMoment,pow_zero,mul_one,lineMoment_X_pow]
lemma X_derivative_X_pow (v : ℕ) :
(X:ℝ[X]) * derivative (X^v) = C (v:ℝ) * X^v := by
cases v with
| zero => simp
| succ v => simp only [derivative_X_pow, Nat.add_sub_cancel]; ring
lemma stein_centered (m h u v : ℕ) :
stein m h (X^v*(X-1)^(u+1)) =
C ((m:ℝ)-h-v-u-1) * (X^v*(X-1)^(u+2)) -
C ((h:ℝ)+v+2*u+2) * (X^v*(X-1)^(u+1)) -
C ((u:ℝ)+1) * (X^v*(X-1)^u) := by
rw [stein_apply,derivative_mul]
have hv := X_derivative_X_pow v
have hd : derivative ((X-1)^(u+1) : ℝ[X]) = C ((u:ℝ)+1)*(X-1)^u := by
simp [derivative_pow,derivative_sub,Nat.cast_add]
rw [hd]
have he : (X:ℝ[X])^2 * (derivative (X^v)*(X-1)^(u+1)) =
C (v:ℝ)*X^(v+1)*(X-1)^(u+1) := by
rw [show (X:ℝ[X])^2 * (derivative (X^v)*(X-1)^(u+1)) =
X * (X*derivative (X^v))*(X-1)^(u+1) from by ring,hv,pow_succ]
ring
rw [mul_add,he]
simp only [map_sub,map_add,map_mul,map_one,C_ofNat,pow_succ]
ring
lemma centeredLineMoment_recurrence {m h u v : ℕ} (hu : h+v+u+1 < m) :
((m:ℝ)-h-v-u-1) * centeredLineMoment m h (u+2) v =
((h:ℝ)+v+2*u+2) * centeredLineMoment m h (u+1) v +
((u:ℝ)+1) * centeredLineMoment m h u v := by
have hp : (X^v*(X-1)^(u+1) : ℝ[X]).natDegree ≤ v+(u+1) := by
apply natDegree_mul_le.trans
simp only [natDegree_pow,natDegree_X,mul_one]
exact Nat.add_le_add_left (by simpa using
(Nat.mul_le_mul_left (u+1) (natDegree_X_sub_C_le (1:ℝ)))) v
have hs := lineMoment_stein m h (X^v*(X-1)^(u+1)) (by omega)
rw [stein_centered] at hs
simp only [LinearMap.map_sub,lineMoment_C_mul] at hs
change _ * centeredLineMoment m h (u+2) v -
_ * centeredLineMoment m h (u+1) v - _ * centeredLineMoment m h u v = 0 at hs
linarith
lemma centeredLineMoment_one {m h v : ℕ} (hv : h+v < m) :
((m:ℝ)-h-v) * centeredLineMoment m h 1 v =
((h:ℝ)+v) * fallingMoment m h v := by
have he : centeredLineMoment m h 1 v =
fallingMoment m h (v+1) - fallingMoment m h v := by
rw [centeredLineMoment,pow_one,mul_sub,mul_one,← pow_succ,map_sub,
lineMoment_X_pow,lineMoment_X_pow]
rw [he]
have hh := fallingMoment_succ hv
linear_combination hh
end BinaryCoordinateSweeps.Sparse
end
end OAI
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