File size: 3,744 Bytes
d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Recursion.UniformProfileGain
namespace OAI
/-! The initial class inherits its lower profile from the actual leaf prefix
counts. The label-selection loss is explicit and paid only once. -/
namespace Falconer
open Set Regularization
attribute [local instance] Classical.propDecidable
theorem block_profile_ratios_from_prefix {N B : ℕ} (hB : 0 < B)
{L : Finset (CellPath 2)} (hL : L ⊆ cubePaths 2 N)
(f : CellPath 2 → List ℕ) {p : CellPath 2} (hp : p ∈ L)
{A D t : ℝ} (hA : 0 ≤ A) (hD : 0 ≤ D)
(hmass : (L.card:ℝ) ≤ D*((classLeaves L (blockLabel L N B f (N/B+1))
(blockLabel L N B f (N/B+1) p)).card:ℝ))
(hprefix : ∀ k ≤ N, ∀ q : CellPath 2,
((prefixFiber L k q).card:ℝ) ≤ A*(2:ℝ)^(-t*(k:ℝ))*(L.card:ℝ)) :
∀ i ≤ N/B+1,
(blockProduct L (blockLabel L N B f (N/B+1)) N B i p:ℝ)/
(blockProduct L (blockLabel L N B f (N/B+1)) N B (N/B+1) p:ℝ) ≤
((2:ℝ)^(N/B+1)*D*A)*(2:ℝ)^(-t*(blockDepth N B i:ℝ)) := by
intro i hi
let K := N/B+1
let F := blockLabel L N B f K
let C := classLeaves L F (F p)
have hpC : p ∈ C := by simp [C,classLeaves,hp]
have hCsub : C ⊆ L := Finset.filter_subset _ _
have hc := blockClass_leaf_count hL B f K p hi (Finset.mem_image.mpr ⟨p,hpC,rfl⟩)
have hr := blockClass_root_count hL hB f hp
apply (div_le_iff₀ (Nat.cast_pos.mpr (blockProduct_pos L F N B K p))).2
calc
(blockProduct L F N B i p:ℝ) ≤ ((prefixFiber C (blockDepth N B i)
(p.take (blockDepth N B i))).card:ℝ) := Nat.cast_le.mpr hc.1
_ ≤ ((prefixFiber L (blockDepth N B i) (p.take (blockDepth N B i))).card:ℝ) :=
Nat.cast_le.mpr (Finset.card_le_card (Finset.filter_subset_filter _ hCsub))
_ ≤ A*(2:ℝ)^(-t*(blockDepth N B i:ℝ))*(L.card:ℝ) :=
hprefix _ (Nat.sub_le N _) _
_ ≤ A*(2:ℝ)^(-t*(blockDepth N B i:ℝ))*(D*(C.card:ℝ)) := by gcongr
_ ≤ A*(2:ℝ)^(-t*(blockDepth N B i:ℝ))*(D*((2:ℝ)^K*
(blockProduct L F N B K p:ℝ))) := by
gcongr
exact_mod_cast hr.2
_ = _ := by ring
theorem exists_initial_profile_class {N B : ℕ} (hB : 0 < B)
{L : Finset (CellPath 2)} (hL : L ⊆ cubePaths 2 N) (hLne : L.Nonempty)
{A t lam : ℝ} (hA : 0 ≤ A)
(hprefix : ∀ k ≤ N, ∀ q : CellPath 2,
((prefixFiber L k q).card:ℝ) ≤ A*(2:ℝ)^(-t*(k:ℝ))*(L.card:ℝ))
(hcost : (2:ℝ)^(N/B+1)*((2*B+1:ℕ):ℝ)^(N/B+1)*A ≤ (2:ℝ)^(lam*(N:ℝ))) :
∃ p ∈ L,
(L.card:ℝ) ≤ ((2*B+1:ℕ):ℝ)^(N/B+1)*
((classLeaves L (blockLabel L N B (fun _ => []) (N/B+1))
(blockLabel L N B (fun _ => []) (N/B+1) p)).card:ℝ) ∧
∀ i ≤ N/B+1,
(blockProduct L (blockLabel L N B (fun _ => []) (N/B+1)) N B i p:ℝ)/
(blockProduct L (blockLabel L N B (fun _ => []) (N/B+1)) N B (N/B+1) p:ℝ) ≤
(2:ℝ)^(lam*(N:ℝ)-t*(blockDepth N B i:ℝ)) := by
obtain ⟨p,hp,hmass⟩ := exists_large_planar_block_class hL hLne B (N/B+1)
have hmassR : (L.card:ℝ) ≤ ((2*B+1:ℕ):ℝ)^(N/B+1)*
((classLeaves L (blockLabel L N B (fun _ => []) (N/B+1))
(blockLabel L N B (fun _ => []) (N/B+1) p)).card:ℝ) := by exact_mod_cast hmass
refine ⟨p,hp,hmassR,?_⟩
have hh := block_profile_ratios_from_prefix hB hL (fun _ => []) hp hA (by positivity) hmassR hprefix
intro i hi
calc
_ ≤ ((2:ℝ)^(N/B+1)*((2*B+1:ℕ):ℝ)^(N/B+1)*A)*
(2:ℝ)^(-t*(blockDepth N B i:ℝ)) := hh i hi
_ ≤ (2:ℝ)^(lam*(N:ℝ))*(2:ℝ)^(-t*(blockDepth N B i:ℝ)) :=
mul_le_mul_of_nonneg_right hcost (by positivity)
_ = _ := by rw [← Real.rpow_add (by norm_num : (0:ℝ) < 2)]; congr 1; ring
end Falconer
end OAI
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