| import OAI.MeasureTheory.Falconer.Estimates.ActualDirectBuffer |
| import OAI.MeasureTheory.Falconer.Recursion.ActualPolicyPayments |
| import OAI.MeasureTheory.Falconer.Recursion.InheritedPolicyState |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
|
|
| theorem remaining_descriptor_companion (A : ℕ → ℝ) {N q a₀ t₀ a : ℕ} |
| (hq : 0 < q) {d : PolicyDescriptor} |
| (hd : d ∈ remainingPolicyDescriptors A N q a₀ t₀ a) : |
| PolicyDescriptor.parallel (max a (descriptorAnchor d-descriptorLength d)) |
| (descriptorAnchor d) ∈ remainingPolicyDescriptors A N q a₀ t₀ a := by |
| classical |
| obtain ⟨hd,hap⟩ := Finset.mem_filter.mp hd |
| obtain ⟨z,hz,hd⟩ := Finset.mem_biUnion.mp hd |
| have hz' := Finset.mem_filter.mp hz |
| have huN : z.2 ≤ N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz'.1).2) |
| have hp := originDescriptor_anchor A hd |
| have hl := originDescriptor_length A hq huN hz'.2.2.2 hd |
| have hLp := originDescriptor_length_le_anchor A hd |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_biUnion.mpr ⟨z,hz,?_⟩,hap⟩ |
| apply Finset.mem_union_left |
| apply Finset.mem_image.mpr |
| refine ⟨max a (descriptorAnchor d-descriptorLength d),Finset.mem_Ico.mpr ⟨?_,?_⟩,?_⟩ |
| · rw [← hp] |
| omega |
| · omega |
| · rw [hp] |
|
|
| structure DirectPolicyWitness (A : ℕ → ℝ) (N q a₀ t₀ a v : ℕ) (sig : ℝ) |
| (d : PolicyDescriptor) : Prop where |
| mem : d ∈ remainingPolicyDescriptors A N q a₀ t₀ a |
| maximal : descriptorLength d = (remainingPolicyDescriptors A N q a₀ t₀ a).sup descriptorLength |
| buffer : sourceFarBuffer N (descriptorLength d) sig = descriptorLength d |
| direct_depth : v ≤ a+2*descriptorLength d |
| anchor_gt : a < descriptorAnchor d |
| anchor_lt : descriptorAnchor d < v |
| width : descriptorAnchor d+10*q ≤ v |
| companion : PolicyDescriptor.parallel (max a (descriptorAnchor d-descriptorLength d)) |
| (descriptorAnchor d) ∈ remainingPolicyDescriptors A N q a₀ t₀ a |
| g_lt : max a (descriptorAnchor d-descriptorLength d) < descriptorAnchor d |
| g_slack : ((max a (descriptorAnchor d-descriptorLength d):ℕ):ℝ)-a ≤ sig*N |
| frequency_slack : 10*sig*N ≤ (v:ℝ)-descriptorAnchor d |
| depth_scale : ((descriptorAnchor d-a:ℕ):ℝ) ≤ (v:ℝ)-descriptorAnchor d+2*sig*N |
| length_scale : (descriptorLength d:ℝ) ≤ (v:ℝ)-descriptorAnchor d+2*sig*N |
| midpoint : 2*descriptorAnchor d ≤ a+v+2*q |
| shrinks : (remainingPolicyDescriptors A N q a₀ t₀ (descriptorAnchor d)).card < |
| (remainingPolicyDescriptors A N q a₀ t₀ a).card |
| payment : potential A a v (10*q)+height A a (descriptorAnchor d) ≤ |
| potential A (descriptorAnchor d) v (10*q) |
|
|
| theorem actual_direct_policy_witness (A : ℕ → ℝ) {N q a₀ t₀ a v : ℕ} {sig : ℝ} |
| (hsig : 0 ≤ sig) (hq : 0 < q) (hceil : ⌈10*sig*(N:ℝ)⌉₊ < q) |
| (hslack : sig*(N:ℝ) ≤ (q:ℝ)) (hslack10 : 10*sig*(N:ℝ) ≤ (q:ℝ)) |
| (ht : (t₀:ℝ) ≤ (v:ℝ)+sig*N) (hw : a+100*q < v) |
| (hdirect : v-2*sourceFarBuffer N |
| ((remainingPolicyDescriptors A N q a₀ t₀ a).sup descriptorLength) sig ≤ a) : |
| ∃ d : PolicyDescriptor, DirectPolicyWitness A N q a₀ t₀ a v sig d := by |
| classical |
| let S := remainingPolicyDescriptors A N q a₀ t₀ a |
| have hb := direct_buffer_eq_length hceil hw hdirect |
| have hS : S.Nonempty := by |
| by_contra h |
| have he := Finset.not_nonempty_iff_eq_empty.mp h |
| have hz : S.sup descriptorLength = 0 := by rw [he]; rfl |
| exact (Nat.ne_of_gt hb.2.2) hz |
| obtain ⟨d,hd,hmax⟩ := Finset.exists_mem_eq_sup S hS descriptorLength |
| dsimp only [S] at hmax |
| have hmax' := hmax.symm |
| have hdir : v ≤ a+2*descriptorLength d := by simpa only [hmax] using hb.2.1 |
| have hmem := Finset.mem_filter.mp hd |
| have hap : a < descriptorAnchor d := hmem.2 |
| obtain ⟨z,hz,hzd⟩ := Finset.mem_biUnion.mp hmem.1 |
| have hz' := Finset.mem_filter.mp hz |
| have huN : z.2 ≤ N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz'.1).2) |
| have hl := originDescriptor_length A hq huN hz'.2.2.2 hzd |
| have hLp := originDescriptor_length_le_anchor A hzd |
| have hpt : descriptorAnchor d+descriptorLength d ≤ t₀ := hl.2.2.trans hz'.2.2.1 |
| have hptR : (descriptorAnchor d:ℝ)+(descriptorLength d:ℝ) ≤ (t₀:ℝ) := by exact_mod_cast hpt |
| have hsigN : 0 ≤ sig*(N:ℝ) := mul_nonneg hsig (Nat.cast_nonneg _) |
| have hdirR : (v:ℝ) ≤ (a:ℝ)+2*(descriptorLength d:ℝ) := by exact_mod_cast hdir |
| have hmidR : 2*(descriptorAnchor d:ℝ) ≤ (a:ℝ)+v+2*q := by linarith |
| have hmid : 2*descriptorAnchor d ≤ a+v+2*q := by exact_mod_cast hmidR |
| have hwidth : descriptorAnchor d+10*q ≤ v := by omega |
| have hwidthR : (descriptorAnchor d:ℝ)+10*(q:ℝ) ≤ (v:ℝ) := by exact_mod_cast hwidth |
| have hn : ((descriptorAnchor d-a:ℕ):ℝ) = (descriptorAnchor d:ℝ)-a := Nat.cast_sub hap.le |
| have hgs : ((descriptorAnchor d-descriptorLength d:ℕ):ℝ) = |
| (descriptorAnchor d:ℝ)-descriptorLength d := Nat.cast_sub hLp |
| refine ⟨d, { |
| mem := hd |
| maximal := hmax' |
| buffer := ?_ |
| direct_depth := hdir |
| anchor_gt := hap |
| anchor_lt := by omega |
| width := hwidth |
| companion := remaining_descriptor_companion A hq hd |
| g_lt := by omega |
| g_slack := ?_ |
| frequency_slack := by linarith |
| depth_scale := ?_ |
| length_scale := ?_ |
| midpoint := hmid |
| shrinks := reached_anchor_strictly_reduces_list A N q a₀ t₀ hap.le hd le_rfl |
| payment := (actual_direct_policy_payment A hq hw hmem.1 hap hdir hmid).2 }⟩ |
| · simpa only [← hmax] using hb.1 |
| · rw [Nat.cast_max,hgs,sub_le_iff_le_add] |
| apply max_le <;> linarith [show 0 ≤ sig*(N:ℝ) by positivity] |
| · rw [hn] |
| linarith |
| · have haR : (a:ℝ) < (descriptorAnchor d:ℝ) := Nat.cast_lt.mpr hap |
| linarith |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
