| import OAI.MeasureTheory.Falconer.Recursion.BranchTransitions | |
| namespace OAI | |
| namespace PlanarFalconer | |
| theorem BranchStep.width_decreases {A : ℕ → ℝ} {N q : ℕ} {sig : ℝ} | |
| (hN : 0 < N) (hq : 0 < q) (hsig : 0 < sig) | |
| {s u : BranchState} {label : BranchLabel} (h : BranchStep A N q sig s label u) | |
| (hs : s.Valid N sig) (hu : u.Valid N sig) : | |
| u.finish-u.start < s.finish-s.start := by | |
| have hn := h.nested hq hs | |
| have hsd := BranchState.valid_depths hs | |
| have hud := BranchState.valid_depths hu | |
| have hstrict : s.start < u.start ∨ u.finish < s.finish := by | |
| cases h with | |
| | suffix hw _ => | |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq hs.2 hw) | |
| exact Or.inr (by dsimp only [BranchState.finish]; omega) | |
| | low _ _ hg => | |
| have hpos : 0 < sig*(N:ℝ) := mul_pos hsig (Nat.cast_pos.mpr hN) | |
| apply Or.inr | |
| apply (Nat.cast_lt (α := ℝ)).mp | |
| dsimp only [BranchState.finish] | |
| linarith | |
| | high hw _ _ _ => | |
| have hp := policyDepth_interior A hq (policyBlocks_nonempty hq hs.2 hw) | |
| exact Or.inl (by dsimp only [BranchState.start]; omega) | |
| | near _ hp => exact Or.inl hp | |
| | far _ _ hm => | |
| have hm := Finset.mem_Icc.mp hm | |
| exact Or.inl (by dsimp only [BranchState.start]; omega) | |
| | direct _ he => exact Or.inl he.anchor_gt | |
| omega | |
| def branchPredecessor (A : ℕ → ℝ) (N q : ℕ) (sig : ℝ) (u s : BranchState) : Prop := | |
| s.Valid N sig ∧ u.Valid N sig ∧ ∃ label, BranchStep A N q sig s label u | |
| theorem branchPredecessor_wellFounded (A : ℕ → ℝ) {N q : ℕ} {sig : ℝ} | |
| (hN : 0 < N) (hq : 0 < q) (hsig : 0 < sig) : | |
| WellFounded (branchPredecessor A N q sig) := by | |
| apply Subrelation.wf (r := (measure (fun s : BranchState => s.finish-s.start)).rel) _ | |
| (measure (fun s : BranchState => s.finish-s.start)).wf | |
| intro u s h | |
| obtain ⟨hs,hu,label,hstep⟩ := h | |
| exact hstep.width_decreases hN hq hsig hs hu | |
| end PlanarFalconer | |
| end OAI | |