| import OAI.MeasureTheory.Falconer.Angular.AngularNearProgress |
| import OAI.MeasureTheory.Falconer.Recursion.BranchRepresentativeCharges |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
|
|
| inductive BranchState where |
| | D (a t i : ℕ) |
| | H (a₀ t₀ a t i : ℕ) |
| deriving DecidableEq |
|
|
| namespace BranchState |
|
|
| def start : BranchState → ℕ |
| | .D a _ _ | .H _ _ a _ _ => a |
| def finish : BranchState → ℕ |
| | .D _ t _ | .H _ _ _ t _ => t |
| def index : BranchState → ℕ |
| | .D _ _ i | .H _ _ _ _ i => i |
| def angular : BranchState → ℕ |
| | .D .. => 0 |
| | .H .. => 1 |
|
|
| def Valid (N : ℕ) (sig : ℝ) : BranchState → Prop |
| | .D a t _ => a ≤ t ∧ t ≤ N |
| | .H a₀ t₀ a t _ => a₀ ≤ a ∧ a ≤ t ∧ t ≤ t₀ ∧ t₀ ≤ N ∧ (t₀:ℝ) ≤ t+sig*N |
|
|
| noncomputable def descriptors (A : ℕ → ℝ) (N q : ℕ) : BranchState → Finset PolicyDescriptor |
| | .D a t _ => policyDescriptorList A N q a t |
| | .H a₀ t₀ a _ _ => remainingPolicyDescriptors A N q a₀ t₀ a |
|
|
| noncomputable def nearDepth (A : ℕ → ℝ) (N q : ℕ) (sig : ℝ) (s : BranchState) : ℕ := |
| s.finish-2*sourceFarBuffer N ((s.descriptors A N q).sup descriptorLength) sig |
|
|
| theorem valid_depths {N : ℕ} {sig : ℝ} {s : BranchState} (h : s.Valid N sig) : |
| s.start ≤ s.finish ∧ s.finish ≤ N := by |
| cases s <;> simp only [Valid,start,finish] at * |
| · exact h |
| · omega |
|
|
| end BranchState |
|
|
| inductive BranchLabel where |
| | suffix | low | high | near | far | direct |
| deriving DecidableEq |
|
|
| inductive BranchStep (A : ℕ → ℝ) (N q : ℕ) (sig : ℝ) : |
| BranchState → BranchLabel → BranchState → Prop where |
| | suffix {a t i : ℕ} (wide : a+100*q < t) |
| (side : a+t < 2*policyDepth A N q a t) : |
| BranchStep A N q sig (.D a t i) .suffix (.D a (policyDepth A N q a t) (i+1)) |
| | low {a t i v : ℕ} (wide : a+100*q < t) |
| (mem : v ∈ Finset.Ioc a t) (gap : sig*N ≤ (t:ℝ)-v) : |
| BranchStep A N q sig (.D a t i) .low (.D a v (i+1)) |
| | high {a t i v : ℕ} (wide : a+100*q < t) |
| (side : 2*policyDepth A N q a t ≤ a+t) |
| (mem : v ∈ Finset.Ioc a t) (gap : (t:ℝ)-v < sig*N) : |
| BranchStep A N q sig (.D a t i) .high (.H a t (policyDepth A N q a t) v i) |
| | near {a₀ t₀ a t i : ℕ} (wide : a+100*q < t) |
| (side : a < (BranchState.H a₀ t₀ a t i).nearDepth A N q sig) : |
| BranchStep A N q sig (.H a₀ t₀ a t i) .near |
| (.H a₀ t₀ ((BranchState.H a₀ t₀ a t i).nearDepth A N q sig) t i) |
| | far {a₀ t₀ a t i b : ℕ} (wide : a+100*q < t) |
| (side : a < (BranchState.H a₀ t₀ a t i).nearDepth A N q sig) |
| (mem : b ∈ Finset.Icc (a+9) ((BranchState.H a₀ t₀ a t i).nearDepth A N q sig+6)) : |
| BranchStep A N q sig (.H a₀ t₀ a t i) .far (.D b t (i+1)) |
| | direct {a₀ t₀ a t i : ℕ} {e : PolicyDescriptor} (wide : a+100*q < t) |
| (witness : DirectPolicyWitness A N q a₀ t₀ a t sig e) : |
| BranchStep A N q sig (.H a₀ t₀ a t i) .direct (.H a₀ t₀ (descriptorAnchor e) t i) |
|
|
| namespace BranchStep |
| variable {A : ℕ → ℝ} {N q : ℕ} {sig : ℝ} |
| variable {s u : BranchState} {label : BranchLabel} |
|
|
| theorem nonterminal (h : BranchStep A N q sig s label u) : s.start+100*q < s.finish := by |
| cases h <;> assumption |
|
|
| theorem nested (h : BranchStep A N q sig s label u) (hq : 0 < q) |
| (hv : s.Valid N sig) : s.start ≤ u.start ∧ u.finish ≤ s.finish := by |
| cases h with |
| | suffix hw _ => |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq hv.2 hw) |
| simp only [BranchState.start,BranchState.finish] |
| omega |
| | low _ hm _ => |
| have hm := Finset.mem_Ioc.mp hm |
| exact ⟨le_rfl,hm.2⟩ |
| | high hw _ hm _ => |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq hv.2 hw) |
| have hm := Finset.mem_Ioc.mp hm |
| exact ⟨by dsimp [BranchState.start]; omega,hm.2⟩ |
| | near _ hs => exact ⟨hs.le,le_rfl⟩ |
| | far _ _ hm => |
| have hm := Finset.mem_Icc.mp hm |
| exact ⟨by dsimp [BranchState.start]; omega,le_rfl⟩ |
| | direct _ hw => exact ⟨hw.anchor_gt.le,le_rfl⟩ |
|
|
| theorem index_change (h : BranchStep A N q sig s label u) : |
| u.index = s.index + (if label = .suffix ∨ label = .low ∨ label = .far then 1 else 0) := by |
| cases h <;> simp [BranchState.index] |
|
|
| theorem episode_balance (h : BranchStep A N q sig s label u) : |
| u.angular + (if label = .far then 1 else 0) = |
| s.angular + (if label = .high then 1 else 0) := by |
| cases h <;> simp [BranchState.angular] |
|
|
| theorem suffix_decrease (h : BranchStep A N q sig s .suffix u) (hq : 0 < q) |
| (hv : s.Valid N sig) : u.finish+q ≤ s.finish := by |
| cases h with |
| | suffix hw _ => |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq hv.2 hw) |
| dsimp only [BranchState.finish] |
| omega |
|
|
| theorem low_decrease (h : BranchStep A N q sig s .low u) : |
| (u.finish:ℝ)+sig*N ≤ s.finish := by |
| cases h with |
| | low _ _ hg => dsimp only [BranchState.finish]; linarith |
|
|
| theorem graph_charge (h : BranchStep A N q sig s label u) (hq : 0 < q) |
| (hv : s.Valid N sig) (hg : label = .high ∨ label = .direct) : |
| s.start < u.start ∧ u.start ∈ policyRepresentatives A N q := by |
| cases h with |
| | high hw _ _ _ => |
| have hn := policyBlocks_nonempty hq hv.2 hw |
| have hp := policyDepth_interior A hq hn |
| exact ⟨by dsimp [BranchState.start]; omega,policyDepth_mem_representatives A hn⟩ |
| | direct _ hw => |
| exact ⟨hw.anchor_gt,policyDescriptor_anchor_mem A hq (Finset.mem_filter.mp hw.mem).1⟩ |
| | _ => simp at hg |
|
|
| end BranchStep |
|
|
| def ChargedNear (A : ℕ → ℝ) (N q : ℕ) (s u : BranchState) : Prop := |
| u.start+100*q < u.finish ∧ ∃ e ∈ s.descriptors A N q, |
| descriptorLength e = (s.descriptors A N q).sup descriptorLength ∧ |
| s.start < descriptorAnchor e ∧ descriptorAnchor e ≤ u.start |
|
|
| theorem near_uncharged_next_direct {A : ℕ → ℝ} {N q : ℕ} {sig : ℝ} |
| (hq : 0 < q) (hceil : ⌈10*sig*(N:ℝ)⌉₊ < q) |
| {s u w : BranchState} {label : BranchLabel} |
| (h : BranchStep A N q sig s .near u) (hn : ¬ ChargedNear A N q s u) |
| (hnext : BranchStep A N q sig u label w) : label = .direct := by |
| have hw := hnext.nonterminal |
| cases h with |
| | near hwide hnear => |
| obtain ht | hp | he := actual_near_progress A hq hceil hnear |
| · exact (not_lt_of_ge ht hw).elim |
| · obtain ⟨e,hm,hL,ha,hb,_⟩ := hp |
| exact (hn ⟨hw,e,hm,hL,ha,hb⟩).elim |
| · simp only [BranchState.finish] at he |
| cases hnext with |
| | near _ hs => |
| simp only [BranchState.nearDepth,BranchState.finish,BranchState.descriptors] at hs |
| rw [he.2] at hs |
| exact (Nat.lt_irrefl _ hs).elim |
| | far _ hs _ => |
| simp only [BranchState.nearDepth,BranchState.finish,BranchState.descriptors] at hs |
| rw [he.2] at hs |
| exact (Nat.lt_irrefl _ hs).elim |
| | direct _ _ => rfl |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
