| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| namespace OAI | |
| namespace PlanarFalconer | |
| noncomputable def remainingPolicyDescriptors (A : ℕ → ℝ) (N q a₀ t₀ a : ℕ) : Finset PolicyDescriptor := | |
| (policyDescriptorList A N q a₀ t₀).filter fun b => a < descriptorAnchor b | |
| theorem policyDescriptor_anchor_gt_left (A : ℕ → ℝ) {N q a t : ℕ} (hq : 0 < q) | |
| {d : PolicyDescriptor} (hd : d ∈ policyDescriptorList A N q a t) : a < descriptorAnchor d := by | |
| obtain ⟨z,hz,hd⟩ := Finset.mem_biUnion.mp hd | |
| obtain ⟨hz,ha,_,hwide⟩ := Finset.mem_filter.mp hz | |
| have huN : z.2 ≤ N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz).2) | |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq huN hwide) | |
| rw [originDescriptor_anchor A hd] | |
| omega | |
| theorem minimal_policy_list_survives (A : ℕ → ℝ) {N q a₀ t₀ a b v : ℕ} | |
| (hq : 0 < q) (hleft : a₀ ≤ b) (hright : v ≤ t₀) (hdepth : a ≤ b) : | |
| policyDescriptorList A N q b v ⊆ remainingPolicyDescriptors A N q a₀ t₀ a := by | |
| intro d hd | |
| exact Finset.mem_filter.mpr ⟨policyDescriptorList_mono A hleft hright hd, | |
| hdepth.trans_lt (policyDescriptor_anchor_gt_left A hq hd)⟩ | |
| theorem remainingPolicyDescriptors_mono (A : ℕ → ℝ) (N q a₀ t₀ : ℕ) {a p : ℕ} (hap : a ≤ p) : | |
| remainingPolicyDescriptors A N q a₀ t₀ p ⊆ remainingPolicyDescriptors A N q a₀ t₀ a := by | |
| intro d hd | |
| obtain ⟨hd,hp⟩ := Finset.mem_filter.mp hd | |
| exact Finset.mem_filter.mpr ⟨hd,hap.trans_lt hp⟩ | |
| theorem remainingPolicyDescriptors_drop (A : ℕ → ℝ) (N q a₀ t₀ : ℕ) {a p : ℕ} (hap : a ≤ p) : | |
| (remainingPolicyDescriptors A N q a₀ t₀ a).filter (fun d => p < descriptorAnchor d) = | |
| remainingPolicyDescriptors A N q a₀ t₀ p := by | |
| ext d | |
| simp only [remainingPolicyDescriptors,Finset.mem_filter] | |
| constructor | |
| · exact fun h => ⟨h.1.1,h.2⟩ | |
| · exact fun h => ⟨⟨h.1,hap.trans_lt h.2⟩,h.2⟩ | |
| theorem remainingPolicyDescriptors_card (A : ℕ → ℝ) {N q a₀ t₀ a : ℕ} (hq : 0 < q) : | |
| (remainingPolicyDescriptors A N q a₀ t₀ a).card ≤ (N+1)^3 := | |
| (Finset.card_filter_le _ _).trans (policyDescriptorList_card A hq) | |
| theorem reached_anchor_strictly_reduces_list (A : ℕ → ℝ) (N q a₀ t₀ : ℕ) {a p : ℕ} | |
| (hap : a ≤ p) {d : PolicyDescriptor} | |
| (hd : d ∈ remainingPolicyDescriptors A N q a₀ t₀ a) (hp : descriptorAnchor d ≤ p) : | |
| (remainingPolicyDescriptors A N q a₀ t₀ p).card < | |
| (remainingPolicyDescriptors A N q a₀ t₀ a).card := by | |
| apply Finset.card_lt_card | |
| apply Finset.ssubset_iff_subset_ne.mpr | |
| refine ⟨remainingPolicyDescriptors_mono A N q a₀ t₀ hap,?_⟩ | |
| intro he | |
| rw [← he] at hd | |
| have hh := (Finset.mem_filter.mp hd).2 | |
| omega | |
| end PlanarFalconer | |
| end OAI | |