| import Mathlib |
| import OAI.Probability.BinarySweep.Trajectories.FallingMoment |
|
|
| namespace OAI |
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|
| 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) |
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| @[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 |
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|
| end |
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| end OAI |
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|
