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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 | 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
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