| 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 | |