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dd14c26 | 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 | import Mathlib
namespace OAI
noncomputable section
namespace ExtremalEdges
def gradientConstant (D : ℕ) : ℝ := 2^(D+1)+4*D+1
end ExtremalEdges
namespace SparseBlocks
structure Parameters where
L : ℕ
even : Even L
δ : ℝ
δ₀ : ℝ
k₀ : ℕ
s₀ : ℝ
δpos : 0 < δ
δle : δ ≤ δ₀
δ₀le : δ₀ ≤ 1/16
klarge : 4/δ₀ ≤ k₀
spos : 0 < s₀
Llarge : 8*((k₀:ℝ)+2)/δ₀ ≤ L
def q (u : ℝ) : ℝ := (ExtremalEdges.gradientConstant 6*u)^(1/ExtremalEdges.gradientConstant 6)
end SparseBlocks
namespace SparseBlocks
open Filter
open scoped Topology
lemma q_root (r : ℝ) (hr : 0 ≤ r) : q (r^153/153) = r := by
have hC : ExtremalEdges.gradientConstant 6 = 153 := by
norm_num [ExtremalEdges.gradientConstant]
rw [q,hC,show (153:ℝ)*(r^153/153) = r^153 by ring,
← Real.rpow_natCast,← Real.rpow_mul hr]
norm_num
lemma exists_window (a b : ℝ) (ha : 0 < a) (hb : 0 < b) :
∃ u : ℝ, 0 < u ∧ u ≤ b ∧ u ≤ 1/8 ∧ q u ≤ a := by
let r := min a (min b (1/8))
have hr : 0 < r := lt_min ha (lt_min hb (by norm_num))
have hra : r ≤ a := min_le_left _ _
have hrb : r ≤ b := (min_le_right _ _).trans (min_le_left _ _)
have hr8 : r ≤ 1/8 := (min_le_right _ _).trans (min_le_right _ _)
have hr1 : r ≤ 1 := hr8.trans (by norm_num)
have hp : r^153 ≤ r := by
simpa only [pow_one] using pow_le_pow_of_le_one hr.le hr1 (show 1 ≤ 153 by norm_num)
have hu : r^153/153 ≤ r := by
apply (div_le_iff₀ (by norm_num : (0:ℝ) < 153)).mpr
nlinarith
exact ⟨r^153/153,div_pos (pow_pos hr _) (by norm_num),hu.trans hrb,
hu.trans hr8,by rw [q_root r hr.le];exact hra⟩
lemma sparse_parameters (η : ℝ) (hη : 0 < η) :
∃ p : Parameters, q (2*p.δ₀) ≤ 1/16 ∧ p.s₀ ≤ 1/8 ∧
((3*p.L+1:ℕ):ℝ)^4*q p.δ ≤ η ∧
((3*p.L+1:ℕ):ℝ)^4*(1/2:ℝ)^p.k₀ ≤ η ∧ 4*q p.s₀ ≤ η := by
obtain ⟨ε,hε,hε8,_,hεq⟩ := exists_window (1/16) (1/8) (by norm_num) (by norm_num)
let δ₀ := ε/2
have hδ₀ : 0 < δ₀ := div_pos hε (by norm_num)
have hδ₀16 : δ₀ ≤ 1/16 := by dsimp [δ₀];linarith
obtain ⟨N,hN⟩ := exists_nat_gt (4/δ₀)
have hscale : 8/δ₀ ≤ 2*(N:ℝ) := by
rw [show 8/δ₀ = 2*(4/δ₀) by ring]
linarith
have hlim : Tendsto (fun k : ℕ => (18*(N:ℝ)+1)^4*((k:ℝ)^4*(1/2:ℝ)^k))
atTop (𝓝 0) := by
simpa using (tendsto_pow_const_mul_const_pow_of_lt_one 4
(by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (1/2:ℝ) < 1)).const_mul ((18*(N:ℝ)+1)^4)
have hsmall := hlim.eventually (gt_mem_nhds hη)
have hlarge : ∀ᶠ k : ℕ in atTop, 4/δ₀ ≤ (k:ℝ) :=
(tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop _)
obtain ⟨k,hk,hk1,hsmallk⟩ := (hlarge.and ((eventually_ge_atTop 1).and hsmall)).exists
let L := 2*N*(k+2)
have hLlarge : 8*((k:ℝ)+2)/δ₀ ≤ L := by
have hh := mul_le_mul_of_nonneg_right hscale (show 0 ≤ (k:ℝ)+2 by positivity)
calc
8*((k:ℝ)+2)/δ₀ = (8/δ₀)*((k:ℝ)+2) := by ring
_ ≤ 2*(N:ℝ)*((k:ℝ)+2) := hh
_ = (L:ℝ) := by dsimp [L];push_cast;ring
have hpoly : ((3*L+1:ℕ):ℝ) ≤ (18*(N:ℝ)+1)*(k:ℝ) := by
dsimp [L]
push_cast
have hk1' : (1:ℝ) ≤ k := by exact_mod_cast hk1
nlinarith [Nat.cast_nonneg (α := ℝ) N]
have hfar : ((3*L+1:ℕ):ℝ)^4*(1/2:ℝ)^k ≤ η := by
calc
_ ≤ ((18*(N:ℝ)+1)*(k:ℝ))^4*(1/2:ℝ)^k :=
mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (by positivity) hpoly 4) (by positivity)
_ = (18*(N:ℝ)+1)^4*((k:ℝ)^4*(1/2:ℝ)^k) := by ring
_ ≤ η := hsmallk.le
have hsize : (0:ℝ) < ((3*L+1:ℕ):ℝ)^4 := by positivity
obtain ⟨δ,hδ,hδle,hδ8,hδq⟩ := exists_window (η/((3*L+1:ℕ):ℝ)^4) δ₀
(div_pos hη hsize) hδ₀
obtain ⟨s,hs,hs8,_,hsq⟩ := exists_window (η/4) (1/8) (by positivity) (by norm_num)
have heven : Even L := ⟨N*(k+2), by dsimp [L]; ring⟩
let p : Parameters := {
L := L
even := heven
δ := δ
δ₀ := δ₀
k₀ := k
s₀ := s
δpos := hδ
δle := hδle
δ₀le := hδ₀16
klarge := hk
spos := hs
Llarge := hLlarge }
refine ⟨p,?_,hs8,?_,hfar,?_⟩
· change q (2*(ε/2)) ≤ 1/16
simpa only [show 2*(ε/2) = ε by ring] using hεq
· change ((3*L+1:ℕ):ℝ)^4*q δ ≤ η
have hh := (le_div_iff₀ hsize).mp hδq
linarith
· change 4*q s ≤ η
linarith
end SparseBlocks
end
end OAI
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