openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /QueryMartingale.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Exploration.ClosedFresh | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace CriticalPercolation.SearchTree | |
| open scoped Classical | |
| open FiniteProduct | |
| variable {E R : Type*} | |
| def queryCount : SearchTree E R → (E → Prop) → ℕ | |
| | .done _, _ => 0 | |
| | .ask e l r, ω => 1 + if ω e then queryCount r ω else queryCount l ω | |
| def centeredSum (p : ℝ) : SearchTree E R → (E → Prop) → ℝ | |
| | .done _, _ => 0 | |
| | .ask e l r, ω => if ω e then (1-p)+centeredSum p r ω else -p+centeredSum p l ω | |
| def truncate : SearchTree E R → ℕ → SearchTree E Unit | |
| | _, 0 => .done () | |
| | .done _, _+1 => .done () | |
| | .ask e l r, n+1 => .ask e (truncate l n) (truncate r n) | |
| lemma truncate_valid [DecidableEq E] (t : SearchTree E R) (n : ℕ) (used : Finset E) | |
| (h : t.Valid used) : (t.truncate n).Valid used := by | |
| induction n generalizing t used with | |
| | zero => simp [truncate,Valid] | |
| | succ n ih => | |
| cases t with | |
| | done r => trivial | |
| | ask e l r => exact ⟨h.1, ih l _ h.2.1, ih r _ h.2.2⟩ | |
| lemma queryCount_truncate_le (t : SearchTree E R) (n : ℕ) (ω : E → Prop) : | |
| (t.truncate n).queryCount ω ≤ n := by | |
| induction n generalizing t with | |
| | zero => simp [truncate,queryCount] | |
| | succ n ih => | |
| cases t with | |
| | done r => simp [truncate,queryCount] | |
| | ask e l r => | |
| simp only [truncate,queryCount] | |
| split_ifs <;> have := ih r <;> have := ih l <;> omega | |
| lemma centeredSum_truncate_eq (t : SearchTree E R) (n : ℕ) (ω : E → Prop) | |
| (h : t.queryCount ω ≤ n) (p : ℝ) : | |
| (t.truncate n).centeredSum p ω = t.centeredSum p ω := by | |
| induction n generalizing t with | |
| | zero => | |
| cases t with | |
| | done r => rfl | |
| | ask e l r => simp [queryCount] at h | |
| | succ n ih => | |
| cases t with | |
| | done r => rfl | |
| | ask e l r => | |
| by_cases he : ω e | |
| · simp only [queryCount,ite_eq_left he] at h | |
| simp only [truncate,centeredSum,ite_eq_left he] | |
| rw [ih r (by omega)] | |
| · simp only [queryCount,ite_eq_right he] at h | |
| simp only [truncate,centeredSum,ite_eq_right he] | |
| rw [ih l (by omega)] | |
| variable [Fintype E] [DecidableEq E] | |
| lemma centeredSum_moments (t : SearchTree E R) (used : Finset E) (ht : t.Valid used) | |
| (p : ℝ) (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) | |
| (hf : LazyDFS.Fresh w p used) : | |
| expect w (t.centeredSum p) = 0 ∧ | |
| expect w (fun ω => (t.centeredSum p ω)^2) = | |
| p*(1-p)*expect w (fun ω => (t.queryCount ω : ℝ)) := by | |
| induction t generalizing used w with | |
| | done r => simp [centeredSum,queryCount,expect_const w hw] | |
| | ask e l r hl hr => | |
| have h0 := hl (insert e used) ht.2.1 (pin w e False) | |
| (pin_normalized w hw e False) (LazyDFS.fresh_pin w p used hf e False) | |
| have h1 := hr (insert e used) ht.2.2 (pin w e True) | |
| (pin_normalized w hw e True) (LazyDFS.fresh_pin w p used hf e True) | |
| have he := hf e ht.1 | |
| constructor | |
| · simp only [centeredSum] | |
| erw [expect_prop_ite] | |
| simp only [expect_add, expect_const _ (pin_normalized w hw e False), | |
| expect_const _ (pin_normalized w hw e True),h0.1,h1.1,he.1,he.2] | |
| ring | |
| · have hx : (fun ω => ((.ask e l r : SearchTree E R).centeredSum p ω)^2) = | |
| (fun ω => if ω e then ((1-p)+r.centeredSum p ω)^2 | |
| else (-p+l.centeredSum p ω)^2) := by funext ω; simp only [centeredSum]; split_ifs <;> rfl | |
| rw [hx] | |
| erw [expect_prop_ite, | |
| expect_square_add_const _ (pin_normalized w hw e False), | |
| expect_square_add_const _ (pin_normalized w hw e True),h0.1,h1.1,h0.2,h1.2,he.1,he.2] | |
| have hc : (fun ω => ((.ask e l r : SearchTree E R).queryCount ω : ℝ)) = | |
| (fun ω => 1+(if ω e then (r.queryCount ω : ℝ) else (l.queryCount ω : ℝ))) := by | |
| funext ω; simp only [queryCount]; split_ifs <;> simp | |
| rw [hc,expect_add] | |
| erw [expect_prop_ite,expect_const w hw,he.1,he.2] | |
| ring | |
| lemma truncated_second_le (t : SearchTree E R) (used : Finset E) (ht : t.Valid used) | |
| (p : ℝ) (hp : p ∈ Set.Icc 0 1) (w : E → Prop → ℝ) | |
| (hw : ∀ e, ∑ b, w e b = 1) (hn : ∀ e b, 0 ≤ w e b) | |
| (hf : LazyDFS.Fresh w p used) (n : ℕ) : | |
| expect w (fun ω => ((t.truncate n).centeredSum p ω)^2) ≤ p*(1-p)*n := by | |
| rw [(t.truncate n).centeredSum_moments used (t.truncate_valid n used ht) p w hw hf |>.2] | |
| apply mul_le_mul_of_nonneg_left _ (mul_nonneg hp.1 (sub_nonneg.mpr hp.2)) | |
| calc expect w (fun ω => ((t.truncate n).queryCount ω : ℝ)) ≤ expect w (fun _ => (n:ℝ)) := | |
| expect_mono hn (fun ω => Nat.cast_le.mpr (t.queryCount_truncate_le n ω)) | |
| _ = n := expect_const w hw _ | |
| end CriticalPercolation.SearchTree | |
| end | |
| end | |
| end | |
| end OAI | |