File size: 8,045 Bytes
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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 | 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
|