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f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.PrefixState
import OAI.Probability.CriticalPercolation.Exploration.ScoreMeanStep
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.LazyDFS
open scoped Classical
open FiniteProduct SearchTree
variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
omit [Fintype V] [Fintype E] in
lemma compile_stop (G : BondGraph V E) (order : List (Dart V E)) (B : Set V)
(fuel : ℕ) (st : State V E) (stack : List (Dart V E))
(hh : ∃ v ∈ st.reached, v ∈ B) :
compile G order B fuel st stack = .done st := by
cases fuel with
| zero => rfl
| succ n => rw [compile.eq_def]; exact ite_eq_left hh
theorem compile_score_second (G : BondGraph V E) (order : List (Dart V E))
(ho : Exhaustive G order) (a t : V) (fuel : ℕ) (st : State V E)
(stack : List (Dart V E)) (hrank : rank order st stack ≤ fuel)
{p : ℝ} (hp : p ∈ Set.Ioo 0 1) (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hn : ∀ e b, 0 ≤ w e b)
(fresh : Fresh w p st.read) (hs : PrefixState G a st w) :
expect w (fun ω => ((compile G order {t} fuel st stack).score p (G.connects a t) w ω)^2) ≤
3*expect w ((compile G order {t} fuel st stack).score p (G.connects a t) w) := by
induction fuel generalizing st stack w with
| zero =>
change expect w (fun _ : E → Prop => (0:ℝ)^2) ≤ 3*expect w (fun _ : E → Prop => (0:ℝ))
simp only [zero_pow (by omega : 2 ≠ 0),expect_const w hw 0,mul_zero,le_refl]
| succ n ih =>
rw [compile.eq_def]
dsimp only
split_ifs with hh
· change expect w (fun _ : E → Prop => (0:ℝ)^2) ≤ 3*expect w (fun _ : E → Prop => (0:ℝ))
simp only [zero_pow (by omega : 2 ≠ 0),expect_const w hw 0,mul_zero,le_refl]
· cases stack with
| nil =>
change expect w (fun _ : E → Prop => (0:ℝ)^2) ≤ 3*expect w (fun _ : E → Prop => (0:ℝ))
simp only [zero_pow (by omega : 2 ≠ 0),expect_const w hw 0,mul_zero,le_refl]
| cons d rest =>
dsimp only
split_ifs with hd
· apply ih st rest (by have := rank_pop order st d rest; omega) w hw hn fresh hs
· have hz : d.2.2 ∉ st.reached := fun hz => hd (Or.inl hz)
have hf : d.2.1 ∉ st.read := fun hf => hd (Or.inr (Or.inl hf))
have hv : d.1 ∈ st.reached := by tauto
have he : G.ends d.2.1 = s(d.1,d.2.2) := by tauto
have ht : t ∉ st.reached := fun ht => hh ⟨t,ht,Set.mem_singleton t⟩
have hb := fresh d.2.1 hf
have h0 : w d.2.1 False ≠ 0 := by rw [hb.1]; exact (sub_pos.mpr hp.2).ne'
have h1 : w d.2.1 True ≠ 0 := by rw [hb.2]; exact hp.1.ne'
have hrc : rank order (st.closed d.2.1) rest ≤ n := by
have := rank_closed order st d rest
omega
have hro : rank order (st.opened d.2.1 d.2.2) (push G order d.2.2++rest) ≤ n := by
have := rank_opened G order st d rest hz
omega
have hrp : rank order (st.opened d.2.1 d.2.2) (push G order d.2.2) ≤ n := by
unfold rank at hro ⊢
simp only [List.length_append] at hro
omega
have hpair : expect (pin w d.2.1 True) (fun ω =>
if openPivot (G.connects a t) d.2.1 ω then
(compile G order {t} n (st.opened d.2.1 d.2.2) (push G order d.2.2++rest)).score p
(G.connects a t) (pin w d.2.1 True) ω else 0) ≤ delta w (G.connects a t) d.2.1 := by
by_cases hzt : d.2.2 = t
· have hstop := compile_stop G order {t} n (st.opened d.2.1 d.2.2)
(push G order d.2.2++rest) ⟨t,by simp [State.opened,hzt],Set.mem_singleton t⟩
simp only [hstop,score,ite_self,expect_const _ (pin_normalized w hw d.2.1 True) 0]
exact (delta_unit w hw hn d.2.1 h0 h1 _ (G.connects_mono a t)).1
· exact future_pair_bound G order ho a t d.1 d.2.2 d.2.1 st rest n
hv hz ht hzt hf he hrp hp w hw hn fresh hs
have hstep := score_second_step d.2.1
(compile G order {t} n (st.closed d.2.1) rest)
(compile G order {t} n (st.opened d.2.1 d.2.2) (push G order d.2.2++rest))
hp w hw hn hb.1 hb.2 (G.connects a t) (G.connects_mono a t) hpair
have hc := ih (st.closed d.2.1) rest hrc (pin w d.2.1 False)
(pin_normalized w hw d.2.1 False) (pin_nonneg w hn d.2.1 False)
(fresh_pin w p st.read fresh d.2.1 False) (prefixState_closed G a st w hs d.2.1 h0)
have ho' := ih (st.opened d.2.1 d.2.2) (push G order d.2.2++rest) hro (pin w d.2.1 True)
(pin_normalized w hw d.2.1 True) (pin_nonneg w hn d.2.1 True)
(fresh_pin w p st.read fresh d.2.1 True) (prefixState_opened G a st w hs d.2.1 d.1 d.2.2 hv he h1)
rw [score_mean_step _ _ _ hp w hw hb.1 hb.2 _ (G.connects_mono a t)]
have hc' := mul_le_mul_of_nonneg_left hc (sub_pos.mpr hp.2).le
have ho'' := mul_le_mul_of_nonneg_left ho' hp.1.le
linarith
end CriticalPercolation.LazyDFS
namespace CriticalPercolation.LazyDFS
open scoped Classical
open FiniteProduct SearchTree
variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
theorem search_pivotal_moments (G : BondGraph V E) (order : List (Dart V E))
(ho : Exhaustive G order) (a t : V) {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
(w : E → Prop → ℝ) (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) :
expect w ((search G order a {t}).score p (G.connects a t) w) ≤ 1 ∧
expect w (fun ω => ((search G order a {t}).score p (G.connects a t) w ω)^2) ≤ 3 := by
have hw : ∀ e, ∑ b, w e b = 1 := by
intro e
simp [Fintype.univ_Prop,h0,h1]
have hn : ∀ e b, 0 ≤ w e b := by
intro e b
by_cases hb : b
· have : b = True := propext (iff_true_intro hb)
rw [this,h1]; exact hp.1.le
· have : b = False := propext (iff_false_intro hb)
rw [this,h0]; exact (sub_pos.mpr hp.2).le
have hf : Fresh w p (initial (E := E) a).read := fun e _ => ⟨h0 e,h1 e⟩
have hm := score_mean_le_variance (search G order a {t}) (initial a).read
(compile_valid G order {t} _ _ _) hp w hw hn hf _ (G.connects_mono a t)
have hmean := hm.trans (variance_le_one w hw hn _)
refine ⟨hmean,?_⟩
have hs := compile_score_second G order ho a t _ (initial a) (push G order a)
le_rfl hp w hw hn hf (prefixState_initial G a w)
change _ ≤ 3*expect w ((search G order a {t}).score p (G.connects a t) w) at hs
exact hs.trans (by linarith)
end CriticalPercolation.LazyDFS
end
end
end OAI
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