File size: 2,752 Bytes
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 62 63 64 65 | 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
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