| 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 |
| |
