| import Mathlib |
| import OAI.Probability.BinarySweep.SparseBounds.CenteredLineMoment |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Sparse |
|
|
| lemma fallingMoment_le_two_pow {m h v : ℕ} (hm : 0 < m) |
| (hl : (h:ℝ)+v ≤ (m:ℝ)/2) : fallingMoment m h v ≤ (2:ℝ)^v := by |
| induction v with |
| | zero => simp |
| | succ v ih => |
| have hmr : (0:ℝ) < m := Nat.cast_pos.mpr hm |
| have hv : h+v < m := by exact_mod_cast (show (h:ℝ)+v < m by push_cast at hl; linarith) |
| have hi := ih (by push_cast at hl; linarith) |
| have hn := (fallingMoment_pos hm (Nat.le_of_lt hv)).le |
| have hn' := (fallingMoment_pos hm (by exact_mod_cast (show (h:ℝ)+(v+1) ≤ m by push_cast at hl; linarith))).le |
| have hs := fallingMoment_succ hv |
| have hd : (m:ℝ)/2 ≤ (m:ℝ)-h-v := by push_cast at hl; linarith |
| have ha := mul_le_mul_of_nonneg_right hd hn' |
| have hb := mul_le_mul_of_nonneg_left hi hmr.le |
| rw [pow_succ] |
| nlinarith |
|
|
| lemma light_real_lt {m h n : ℕ} {η : ℝ} (hm : 0 < m) (hη : η ≤ 1/4) |
| (hl : (h:ℝ)+n ≤ m*η) : h+n < m := by |
| have hp : (0:ℝ) < m := Nat.cast_pos.mpr hm |
| exact_mod_cast (show (h:ℝ)+n < m by nlinarith) |
|
|
| theorem centeredLineMoment_light {m h u v : ℕ} {η : ℝ} |
| (hm : 0 < m) (hη0 : 0 ≤ η) (hη : η ≤ 1/4) |
| (hl : (h:ℝ)+u+v ≤ m*η) : |
| 0 ≤ centeredLineMoment m h u v ∧ |
| centeredLineMoment m h u v ≤ (2:ℝ)^v * (4*Real.sqrt η)^u := by |
| have hmr : (0:ℝ) < m := Nat.cast_pos.mpr hm |
| have hr0 : 0 ≤ 4*Real.sqrt η := by positivity |
| have hr2 : 4*Real.sqrt η ≤ 2 := by |
| have hs := Real.sq_sqrt hη0 |
| have hs0 := Real.sqrt_nonneg η |
| nlinarith |
| have hrsq : (4*Real.sqrt η)^2 = 16*η := by rw [mul_pow,Real.sq_sqrt hη0]; norm_num |
| have hetr : η ≤ 4*Real.sqrt η := by nlinarith [Real.sq_sqrt hη0,Real.sqrt_nonneg η] |
| induction u using Nat.strong_induction_on with |
| | h u ih => |
| cases u with |
| | zero => |
| rw [centeredLineMoment_zero,pow_zero,mul_one] |
| have hv : h+v ≤ m := Nat.le_of_lt (light_real_lt hm hη (by simpa using hl)) |
| exact ⟨(fallingMoment_pos hm hv).le,fallingMoment_le_two_pow hm (by nlinarith)⟩ |
| | succ u => |
| cases u with |
| | zero => |
| have hb := fallingMoment_le_two_pow hm (show (h:ℝ)+v ≤ m/2 by push_cast at hl; nlinarith) |
| have hv : h+v < m := light_real_lt hm hη (by push_cast at hl; linarith) |
| have hbn := (fallingMoment_pos hm (Nat.le_of_lt hv)).le |
| have he := centeredLineMoment_one hv |
| have hd : 0 < (m:ℝ)-h-v := by |
| have hv' : (h:ℝ)+v < m := by exact_mod_cast hv |
| linarith |
| have hp : 0 ≤ centeredLineMoment m h 1 v := by |
| apply nonneg_of_mul_nonneg_right (by rw [he]; positivity) hd |
| refine ⟨hp,?_⟩ |
| rw [pow_one] |
| have ha : (m:ℝ)/2 ≤ (m:ℝ)-h-v := by push_cast at hl; nlinarith |
| have hc : (h:ℝ)+v ≤ m*η := by push_cast at hl; linarith |
| have hb' := mul_le_mul_of_nonneg_left hb (show 0 ≤ (h:ℝ)+v by positivity) |
| have hc' := mul_le_mul_of_nonneg_right hc (show (0:ℝ) ≤ 2^v by positivity) |
| have ha' := mul_le_mul_of_nonneg_right ha hp |
| have hr : 2*η ≤ 4*Real.sqrt η := by nlinarith [Real.sq_sqrt hη0,Real.sqrt_nonneg η] |
| have hr' := mul_le_mul_of_nonneg_left hr (show 0 ≤ (m:ℝ)*(2:ℝ)^v by positivity) |
| nlinarith |
| | succ u => |
| have hu1 := ih (u+1) (by omega) (by push_cast at hl ⊢; linarith) |
| have hu0 := ih u (by omega) (by push_cast at hl ⊢; linarith) |
| have hv : h+v+u+1 < m := by |
| have hh : (h:ℝ)+(v+u+1) ≤ m*η := by push_cast at hl; linarith |
| have ht := light_real_lt (h := h) (n := v+u+1) hm hη (by simpa only [Nat.cast_add,Nat.cast_one] using hh) |
| omega |
| have he := centeredLineMoment_recurrence hv |
| have hd : 0 < (m:ℝ)-h-v-u-1 := by |
| have hv' : (h:ℝ)+v+u+1 < m := by exact_mod_cast hv |
| linarith |
| have hp : 0 ≤ centeredLineMoment m h (u+2) v := by |
| apply nonneg_of_mul_nonneg_right (by |
| rw [he] |
| exact add_nonneg (mul_nonneg (by positivity) hu1.1) |
| (mul_nonneg (by positivity) hu0.1)) hd |
| refine ⟨hp,?_⟩ |
| have ha : (m:ℝ)/2 ≤ (m:ℝ)-h-v-u-1 := by push_cast at hl; nlinarith |
| have hc : (h:ℝ)+v+2*u+2 ≤ 2*m*η := by |
| have hhv : (0:ℝ) ≤ h+v := by positivity |
| push_cast at hl |
| linarith |
| have hc0 : 0 ≤ (h:ℝ)+v+2*u+2 := by positivity |
| have hb : (u:ℝ)+1 ≤ m*η := by |
| have hhv : (0:ℝ) ≤ h+v := by positivity |
| push_cast at hl |
| linarith |
| have ha' := mul_le_mul_of_nonneg_right ha hp |
| have hc' := mul_le_mul hc hu1.2 hu1.1 (by positivity : 0 ≤ 2*(m:ℝ)*η) |
| have hb' := mul_le_mul hb hu0.2 hu0.1 (by positivity : 0 ≤ (m:ℝ)*η) |
| have hr : 4*η*(4*Real.sqrt η)+2*η ≤ (4*Real.sqrt η)^2 := by |
| rw [hrsq] |
| nlinarith |
| have hr' := mul_le_mul_of_nonneg_left hr |
| (show 0 ≤ (m:ℝ)/2*((2:ℝ)^v*(4*Real.sqrt η)^u) by positivity) |
| simp only [pow_succ] at hc' ⊢ |
| nlinarith |
|
|
| end BinaryCoordinateSweeps.Sparse |
|
|
| end |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Sparse |
|
|
| variable {L : Type*} [Fintype L] |
|
|
| def multilineMoment (m h : L → ℕ) : MvPolynomial L ℝ →ₗ[ℝ] ℝ := |
| (MvPolynomial.basisMonomials L ℝ).constr ℝ |
| (fun n => ∏ l, fallingMoment (m l) (h l) (n l)) |
|
|
| lemma multilineMoment_monomial (m h : L → ℕ) (n : L →₀ ℕ) : |
| multilineMoment m h (MvPolynomial.monomial n 1) = |
| ∏ l, fallingMoment (m l) (h l) (n l) := by |
| exact (MvPolynomial.basisMonomials L ℝ).constr_basis ℝ _ n |
|
|
| lemma multiline_prod_X (n : L → ℕ) : |
| (∏ l, (MvPolynomial.X l : MvPolynomial L ℝ) ^ n l) = |
| MvPolynomial.monomial (Finsupp.equivFunOnFinite.symm n) 1 := by |
| rw [← MvPolynomial.prod_X_pow_eq_monomial] |
| change (∏ l, MvPolynomial.X l ^ (Finsupp.equivFunOnFinite.symm n) l) = _ |
| symm |
| apply Finset.prod_subset (Finset.subset_univ _) |
| intro l _ hl |
| have hz := Finsupp.notMem_support_iff.mp hl |
| change n l = 0 at hz |
| change (MvPolynomial.X l : MvPolynomial L ℝ)^n l = 1 |
| rw [hz,pow_zero] |
|
|
| theorem multilineMoment_factor (m h : L → ℕ) (P : L → Polynomial ℝ) : |
| multilineMoment m h (∏ l, Polynomial.aeval (MvPolynomial.X l) (P l)) = |
| ∏ l, lineMoment (m l) (h l) (P l) := by |
| let f : MultilinearMap ℝ (fun _ : L => Polynomial ℝ) ℝ := |
| (multilineMoment m h).compMultilinearMap |
| ((MultilinearMap.mkPiAlgebra ℝ L (MvPolynomial L ℝ)).compLinearMap |
| (fun l => (Polynomial.aeval (MvPolynomial.X l)).toLinearMap)) |
| let g : MultilinearMap ℝ (fun _ : L => Polynomial ℝ) ℝ := |
| (MultilinearMap.mkPiAlgebra ℝ L ℝ).compLinearMap (fun l => lineMoment (m l) (h l)) |
| have he : f = g := by |
| apply Module.Basis.ext_multilinear (fun _ : L => Polynomial.basisMonomials ℝ) |
| intro n |
| change multilineMoment m h (∏ l, Polynomial.aeval (MvPolynomial.X l) |
| (Polynomial.basisMonomials ℝ (n l))) = |
| ∏ l, lineMoment (m l) (h l) (Polynomial.basisMonomials ℝ (n l)) |
| simp only [Polynomial.coe_basisMonomials,Polynomial.aeval_monomial,map_one,one_mul, |
| lineMoment_monomial] |
| rw [multiline_prod_X,multilineMoment_monomial] |
| rfl |
| exact congrArg (fun a => a P) he |
|
|
| lemma multilineMoment_centered (m h u v : L → ℕ) : |
| multilineMoment m h (∏ l, (MvPolynomial.X l : MvPolynomial L ℝ)^v l * |
| (MvPolynomial.X l-1)^u l) = |
| ∏ l, centeredLineMoment (m l) (h l) (u l) (v l) := by |
| change multilineMoment m h _ = ∏ l, lineMoment (m l) (h l) |
| (Polynomial.X^v l * (Polynomial.X-1)^u l) |
| rw [← multilineMoment_factor] |
| apply congrArg (multilineMoment m h) |
| apply Finset.prod_congr rfl |
| intro l _ |
| rw [map_mul,map_pow,map_pow,map_sub,map_one,Polynomial.aeval_X] |
|
|
| end BinaryCoordinateSweeps.Sparse |
|
|
| end |
|
|
| end OAI |
|
|
