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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 | 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
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