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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Dynamics.Crossing
namespace OAI
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
structure GreedyRecord (n : ℕ) where
refs : ℕ → Config n
time : ℕ
kept : Finset ℕ
def greedyRun {n : ℕ} (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n)
(x : ℕ → Config n) (r : ℕ → ℝ) (b : ℝ) (m : ℕ) : ℕ → GreedyRecord n
| 0 => ⟨base,0,∅⟩
| j+1 =>
let s := greedyRun base pick x r b m j
let w := pick j s.refs s.time
⟨Function.update s.refs j w,
if j=0 then firstCrossing (fun k => overlap w (x k)) (r j) m
else lastCrossing (fun k => overlap w (x k)) (r j) s.time,
insert j (s.kept\s.kept.filter (fun i => b < |overlap (s.refs i) w|))⟩
def greedyReference {n : ℕ} (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n)
(x : ℕ → Config n) (r : ℕ → ℝ) (b : ℝ) (m j : ℕ) : Config n :=
pick j (greedyRun base pick x r b m j).refs (greedyRun base pick x r b m j).time
theorem greedyRun_predictable {n : ℕ} (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n)
(x : ℕ → Config n) (r s : ℕ → ℝ) (b : ℝ) (m j : ℕ)
(h : ∀ i < j, r i=s i) : greedyRun base pick x r b m j=greedyRun base pick x s b m j := by
induction j with
| zero => rfl
| succ j ih =>
have hp := ih (fun i hi => h i (by omega))
simp only [greedyRun,hp,h j (by omega)]
theorem greedyReference_predictable {n : ℕ} (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n)
(x : ℕ → Config n) (r s : ℕ → ℝ) (b : ℝ) (m j : ℕ)
(h : ∀ i < j, r i=s i) : greedyReference base pick x r b m j=greedyReference base pick x s b m j := by
unfold greedyReference
rw [greedyRun_predictable base pick x r s b m j h]
theorem greedyRun_refs {n : ℕ} (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n)
(x : ℕ → Config n) (r : ℕ → ℝ) (b : ℝ) (m j i : ℕ) (hi : i < j) :
(greedyRun base pick x r b m j).refs i=greedyReference base pick x r b m i := by
induction j with
| zero => omega
| succ j ih =>
by_cases he : i=j
· subst i
simp only [greedyRun,Function.update_self]
rfl
· simpa only [greedyRun,Function.update_of_ne he] using ih (by omega)
theorem greedyRun_range {n : ℕ} (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n)
(x : ℕ → Config n) (r : ℕ → ℝ) (b : ℝ) (m j : ℕ) :
∀ i ∈ (greedyRun base pick x r b m j).kept, i < j := by
induction j with
| zero => simp [greedyRun]
| succ j ih =>
intro i hi
simp only [greedyRun,Finset.mem_insert] at hi
rcases hi with rfl | hi
· omega
· have := ih i (Finset.mem_sdiff.mp hi).1
omega
def bankPick {n : ℕ} (pool : ℕ → Finset (Config n)) (fallback : ℕ → Config n)
(B : ℕ) (x : ℕ → Config n) (t q : ℝ)
(j : ℕ) (v : ℕ → Config n) (u : ℕ) : Config n :=
if h : ∃ w ∈ pool j, q ≤ overlap w (x u) ∧
∀ i, j/B*B ≤ i → i < j → |overlap (v i) w| ≤ 3*t
then h.choose else fallback j
theorem bankPick_mem {n : ℕ} (pool : ℕ → Finset (Config n)) (fallback : ℕ → Config n)
(hf : ∀ j, fallback j ∈ pool j) (B : ℕ) (x : ℕ → Config n) (t q : ℝ)
(j : ℕ) (v : ℕ → Config n) (u : ℕ) : bankPick pool fallback B x t q j v u ∈ pool j := by
unfold bankPick
split_ifs with h
· exact h.choose_spec.1
· exact hf j
theorem bankPick_spec {n : ℕ} (pool : ℕ → Finset (Config n)) (fallback : ℕ → Config n)
(B : ℕ) (x : ℕ → Config n) (t q : ℝ)
(j : ℕ) (v : ℕ → Config n) (u : ℕ)
(h : ∃ w ∈ pool j, q ≤ overlap w (x u) ∧
∀ i, j/B*B ≤ i → i < j → |overlap (v i) w| ≤ 3*t) :
q ≤ overlap (bankPick pool fallback B x t q j v u) (x u) ∧
∀ i, j/B*B ≤ i → i < j → |overlap (v i) (bankPick pool fallback B x t q j v u)| ≤ 3*t := by
rw [bankPick,dite_eq_left h]
exact h.choose_spec.2
end SK.Analytic
end
end
end OAI
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