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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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 | 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
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