| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler |
| open scoped Classical |
|
|
| inductive RoutineKind where |
| | mean | sample |
| deriving DecidableEq |
|
|
| structure RoutineLabel where |
| kind : RoutineKind |
| b : ℝ |
| p : ℝ |
|
|
| def RoutineLabel.terminal (L : RoutineLabel) : Prop := L.p≤L.b |
|
|
| inductive RoutineStep (t : ℝ) : RoutineLabel → RoutineLabel → Prop where |
| | meanFinal (b p : ℝ) : RoutineStep t ⟨.mean,b,p⟩ ⟨.mean,b+t,p⟩ |
| | sampleFinal (b p : ℝ) : RoutineStep t ⟨.sample,b,p⟩ ⟨.mean,b,p⟩ |
| | internal (k k' : RoutineKind) (b p : ℝ) : |
| RoutineStep t ⟨k,b,p⟩ ⟨k',b-10*t,p-1/4⟩ |
|
|
| def RoutineLabel.rank (t : ℝ) (L : RoutineLabel) : ℕ := |
| 2*⌈(L.p-L.b)/t⌉₊+(match L.kind with | .mean => 0 | .sample => 1) |
|
|
| lemma ceil_gap_drop {t x y : ℝ} (ht : 0<t) (hx : 0<x) (hy : y≤x-t) : |
| ⌈y/t⌉₊<⌈x/t⌉₊ := by |
| have hp : 0<x/t := div_pos hx ht |
| have hxy : y/t≤x/t-1 := by |
| simpa only [sub_div,div_self ht.ne'] using (div_le_div_of_nonneg_right hy ht.le) |
| have hn : 0<⌈x/t⌉₊ := Nat.ceil_pos.mpr hp |
| have hc := Nat.le_ceil (x/t) |
| have hm : y/t≤(⌈x/t⌉₊-1:ℕ) := by |
| rw [Nat.cast_sub (by omega : 1≤⌈x/t⌉₊),Nat.cast_one] |
| linarith |
| exact lt_of_le_of_lt (Nat.ceil_le.mpr hm) (Nat.sub_lt hn (by omega)) |
|
|
| lemma RoutineStep.rank_lt {t : ℝ} (ht : 0<t) (hs : t≤1/44) |
| {P C : RoutineLabel} (hP : ¬P.terminal) (h : RoutineStep t P C) : |
| C.rank t<P.rank t := by |
| cases h with |
| | meanFinal b p => |
| have hd := ceil_gap_drop ht (show 0<p-b by simpa [RoutineLabel.terminal] using hP) |
| (y:=p-(b+t)) (by linarith) |
| simp only [RoutineLabel.rank] |
| omega |
| | sampleFinal b p => simp [RoutineLabel.rank] |
| | internal k k' b p => |
| have hd := ceil_gap_drop ht (show 0<p-b by simpa [RoutineLabel.terminal] using hP) |
| (y:=p-1/4-(b-10*t)) (by linarith) |
| cases k <;> cases k' <;> simp only [RoutineLabel.rank] <;> omega |
|
|
| theorem routine_induction {t : ℝ} (ht : 0<t) (hs : t≤1/44) |
| (P : RoutineLabel → Prop) (hbase : ∀L,L.terminal → P L) |
| (hstep : ∀L,¬L.terminal → (∀C,RoutineStep t L C → P C) → P L) : ∀L,P L := by |
| intro L |
| induction hn : L.rank t using Nat.strong_induction_on generalizing L with |
| | h n ih => |
| by_cases hL : L.terminal |
| · exact hbase L hL |
| · apply hstep L hL |
| intro C hc |
| exact ih _ (hn ▸ hc.rank_lt ht hs hL) C rfl |
| end LogConcaveSampling.OracleCompiler |
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| end |
|
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| end |
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| end OAI |
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|
