| import OAI.MeasureTheory.Falconer.Recursion.BranchTransitions | |
| namespace OAI | |
| namespace PlanarFalconer | |
| theorem high_depth_gap {N q t v : ℕ} {sig : ℝ} (hv : v ≤ t) | |
| (hgap : (t:ℝ)-v < sig*N) (hslack : sig*(N:ℝ) ≤ q) : t-v ≤ q := by | |
| have hcast : ((t-v:ℕ):ℝ) = (t:ℝ)-v := Nat.cast_sub hv | |
| exact Nat.cast_le.mp (by rw [hcast]; exact hgap.le.trans hslack) | |
| theorem BranchStep.valid_successor {A : ℕ → ℝ} {N q : ℕ} {sig : ℝ} | |
| (hq : 0 < q) (hslack : sig*(N:ℝ) ≤ q) | |
| (hbuffer : ∀ L t a : ℕ, a < t-2*sourceFarBuffer N L sig → | |
| t-2*sourceFarBuffer N L sig+5 < t) | |
| {s u : BranchState} {label : BranchLabel} | |
| (h : BranchStep A N q sig s label u) (hv : s.Valid N sig) : u.Valid N sig := by | |
| cases h with | |
| | suffix hw _ => | |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq hv.2 hw) | |
| dsimp only [BranchState.Valid] at * | |
| omega | |
| | low _ hm _ => | |
| have hm := Finset.mem_Ioc.mp hm | |
| dsimp only [BranchState.Valid] at * | |
| omega | |
| | high hw hs hm hg => | |
| have hm := Finset.mem_Ioc.mp hm | |
| have hgap := high_depth_gap hm.2 hg hslack | |
| have hp := actual_prefix_policy_payment A hq hv.2 hw hs hm.2 hgap | |
| have hi := policyDepth_interior A hq (policyBlocks_nonempty hq hv.2 hw) | |
| exact ⟨by omega,by omega,hm.2,hv.2,by linarith⟩ | |
| | near _ hs => | |
| dsimp only [BranchState.Valid,BranchState.nearDepth,BranchState.finish,BranchState.descriptors] at * | |
| exact ⟨by omega,Nat.sub_le _ _,hv.2.2⟩ | |
| | far _ hs hm => | |
| have hb := hbuffer _ _ _ hs | |
| have hm := Finset.mem_Icc.mp hm | |
| dsimp only [BranchState.Valid,BranchState.nearDepth,BranchState.finish,BranchState.descriptors] at * | |
| exact ⟨by omega,by omega⟩ | |
| | direct _ hw => | |
| exact ⟨hv.1.trans hw.anchor_gt.le,hw.anchor_lt.le,hv.2.2⟩ | |
| theorem uniform_branch_successor_validity {sig kap : ℝ} | |
| (hsig : 0 < sig) (hkap : 0 < kap) (hgap : sig < kap/1000) : | |
| ∃ N₀ : ℕ, 0 < N₀ ∧ ∀ N ≥ N₀, ∀ A : ℕ → ℝ, | |
| ∀ s u : BranchState, ∀ label : BranchLabel, | |
| s.Valid N sig → BranchStep A N (sourcePolicyBlock N kap) sig s label u → | |
| u.Valid N sig ∧ s.start ≤ u.start ∧ u.finish ≤ s.finish := by | |
| obtain ⟨N₁,hN₁,h₁⟩ := uniform_source_ceiling_slack hsig hgap | |
| obtain ⟨N₂,h₂⟩ := uniform_source_far_buffer hsig | |
| refine ⟨max N₁ N₂,by omega,?_⟩ | |
| intro N hN A s u label hv h | |
| have hq := sourcePolicyBlock_pos (by omega : 0 < N) hkap | |
| have hb : ∀ L t a : ℕ, a < t-2*sourceFarBuffer N L sig → | |
| t-2*sourceFarBuffer N L sig+5 < t := by | |
| intro L t a ha | |
| have hh := h₂ N (by omega) L t a ha | |
| omega | |
| exact ⟨h.valid_successor hq (h₁ N (by omega)).2.2 hb hv,h.nested hq hv⟩ | |
| end PlanarFalconer | |
| end OAI | |