| 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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|
