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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 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 | import Mathlib
import OAI.Probability.BinarySweep.SparseBounds.SparseNumerics
namespace OAI
noncomputable section
open scoped BigOperators
namespace BinaryCoordinateSweeps.Sparse
lemma sparse_amplitudes_small (b h k : ℕ) {L : ℝ} (hb : b≤16000)
(hh : (h:ℝ)≤100000*k) (hL : (10:ℝ)^26≤L) :
((2:ℝ)^(k+k*b)*Real.exp ((b*h+k+k*b:ℕ):ℝ))^2≤Real.exp ((k:ℝ)*L/2000000000) ∧
((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2≤Real.exp ((k:ℝ)*L/2000000000) ∧
(2:ℝ)^k≤Real.exp ((k:ℝ)*L/2000000000) := by
have hbR : (b:ℝ)≤16000 := by exact_mod_cast hb
have h2 : Real.log 2≤2 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)).trans (by norm_num)
have h4 : Real.log 4≤4 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<4)).trans (by norm_num)
have hbK := mul_le_mul_of_nonneg_right hbR (show (0:ℝ)≤k by positivity)
have hhB := mul_le_mul_of_nonneg_left hh (show (0:ℝ)≤b by positivity)
have hLK := mul_le_mul_of_nonneg_right hL (show (0:ℝ)≤k by positivity)
have hp (n : ℕ) : (2:ℝ)^n=Real.exp ((n:ℝ)*Real.log 2) := by
rw [Real.exp_nat_mul,Real.exp_log (by norm_num)]
constructor
· rw [hp,←Real.exp_add,←Real.exp_nat_mul]
apply Real.exp_le_exp.mpr
have h2K := mul_le_mul_of_nonneg_left h2 (show (0:ℝ)≤(k+k*b:ℕ) by positivity)
push_cast at h2K ⊢
nlinarith only [h2K,hhB,hbK,hLK]
constructor
· rw [hp,←Real.exp_add,←Real.exp_nat_mul]
apply Real.exp_le_exp.mpr
have h2K := mul_le_mul_of_nonneg_left h2 (show (0:ℝ)≤k by positivity)
have h4K := mul_le_mul_of_nonneg_right h4 (show (0:ℝ)≤k*b by positivity)
norm_num
nlinarith only [h2K,h4K,hhB,hbK,hLK]
· rw [hp]
apply Real.exp_le_exp.mpr
have h2K := mul_le_mul_of_nonneg_left h2 (show (0:ℝ)≤k by positivity)
nlinarith only [h2K,hLK]
lemma few_hs_scalar (b h k wv wl : ℕ) {L g v x y : ℝ} (C : Fin b → ℝ)
(hb : b≤16000) (hk : 1≤k) (hh : (h:ℝ)≤100000*k) (hL : (10:ℝ)^26≤L)
(htv : (k:ℝ)≤8*(((wv+1)/2:ℕ):ℝ)) (htl : (k:ℝ)≤128000*wl)
(hg0 : 0≤g) (hg : g≤Real.exp (-L/640000))
(hv0 : 0≤v) (hv : v≤Real.exp (-L/3200))
(hx0 : 0≤x) (hx : x≤Real.exp (-L/3200))
(hy0 : 0≤y) (hy : y≤Real.exp (-L/3200))
(hC0 : ∀j,0≤C j) (hC : ∀j,C j≤Real.exp ((k:ℝ)*L/2000000000)) :
((2:ℝ)^(k+k*b)*Real.exp ((b*h+k+k*b:ℕ):ℝ)*g^((k+1)/2))^2+
((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 *
((2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl))≤
Real.exp (-(k:ℝ)*L/1000000000000) := by
have hL0 : 0≤L := (by positivity : (0:ℝ)≤10^26).trans hL
have hkR : (1:ℝ)≤k := by exact_mod_cast hk
have hbR : (b:ℝ)≤16000 := by exact_mod_cast hb
have hKL := mul_nonneg (Nat.cast_nonneg k) hL0
have hkt : (k:ℝ)≤2*(((k+1)/2:ℕ):ℝ) := by exact_mod_cast (show k≤2*((k+1)/2) from by omega)
obtain ⟨hA,hB,h2⟩ := sparse_amplitudes_small b h k hb hh hL
have hgpow : (g^((k+1)/2))^2≤Real.exp (-(k:ℝ)*L/1000000000) := by
have he := nat_pow_le_exp_of_le hg0 hg ((k+1)/2)
have he2 := pow_le_pow_left₀ (pow_nonneg hg0 _) he 2
rw [←Real.exp_nat_mul] at he2
refine he2.trans (Real.exp_le_exp.mpr ?_)
have ht := mul_le_mul_of_nonneg_right hkt hL0
norm_num
nlinarith only [ht,hKL]
have hpow (a : ℝ) (ha0 : 0≤a) (ha : a≤Real.exp (-L/3200))
(t : ℕ) (ht : (k:ℝ)≤128000*t) :
a^t≤Real.exp (-2*(k:ℝ)*L/1000000000) := by
apply (nat_pow_le_exp_of_le ha0 ha t).trans (Real.exp_le_exp.mpr ?_)
have he := mul_le_mul_of_nonneg_right ht hL0
nlinarith only [he,hKL]
have hlx := hpow x hx0 hx wl htl
have hly := hpow y hy0 hy wl htl
have hlv := hpow v hv0 hv ((wv+1)/2) (htv.trans (by nlinarith only [show (0:ℝ)≤(((wv+1)/2:ℕ):ℝ) by positivity]))
have hgood : ((2:ℝ)^(k+k*b)*Real.exp ((b*h+k+k*b:ℕ):ℝ)*g^((k+1)/2))^2 ≤
Real.exp (-(k:ℝ)*L/2000000000) := by
rw [mul_pow]
refine (mul_le_mul hA hgpow (by positivity) (Real.exp_pos _).le).trans_eq ?_
rw [←Real.exp_add]
congr 1
ring
have hvertex : (2:ℝ)^k*v^((wv+1)/2)≤Real.exp (-(k:ℝ)*L/1000000000) := by
refine (mul_le_mul h2 hlv (by positivity) (Real.exp_pos _).le).trans ?_
rw [←Real.exp_add]
apply Real.exp_le_exp.mpr
nlinarith only [hKL]
have hholes : (2:ℝ)^k*x^wl≤Real.exp (-(k:ℝ)*L/1000000000) := by
refine (mul_le_mul h2 hlx (by positivity) (Real.exp_pos _).le).trans ?_
rw [←Real.exp_add]
apply Real.exp_le_exp.mpr
nlinarith only [hKL]
have hparticles (j : Fin b) : C j*(2:ℝ)^k*y^wl≤Real.exp (-(k:ℝ)*L/1000000000) := by
have hc0 := hC0 j
refine (mul_le_mul (mul_le_mul (hC j) h2 (by positivity) (Real.exp_pos _).le)
hly (by positivity) (by positivity)).trans_eq ?_
rw [←Real.exp_add,←Real.exp_add]
congr 1
ring
have hevents : (2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl) ≤
(1+2*b)*Real.exp (-(k:ℝ)*L/1000000000) := by
refine (add_le_add hvertex (Finset.sum_le_sum (fun j _ => add_le_add hholes (hparticles j)))).trans_eq ?_
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
ring
have hbad : ((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 *
((2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl)) ≤
(1+2*b)*Real.exp (-(k:ℝ)*L/2000000000) := by
have hev0 : 0≤ (2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl) := by
apply add_nonneg (by positivity)
apply Finset.sum_nonneg
intro j _
have hc0 := hC0 j
positivity
refine (mul_le_mul hB hevents hev0 (Real.exp_pos _).le).trans_eq ?_
rw [mul_left_comm,←Real.exp_add]
congr 1
ring_nf
have hcount : 2+2*(b:ℝ)≤Real.exp ((k:ℝ)*L/4000000000) := by
apply (show 2+2*(b:ℝ)≤(k:ℝ)*L/4000000000+1 from ?_).trans (Real.add_one_le_exp _)
have hh := mul_le_mul_of_nonneg_right hkR hL0
nlinarith only [hh,hL,hbR]
calc
_ ≤ (2+2*b)*Real.exp (-(k:ℝ)*L/2000000000) := by nlinarith only [hgood,hbad]
_ ≤ Real.exp ((k:ℝ)*L/4000000000)*Real.exp (-(k:ℝ)*L/2000000000) :=
mul_le_mul_of_nonneg_right hcount (Real.exp_pos _).le
_ ≤ _ := by
rw [←Real.exp_add]
apply Real.exp_le_exp.mpr
nlinarith only [hKL]
end BinaryCoordinateSweeps.Sparse
end
end OAI
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