openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Storage /DecisionRepetitionCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.DecisionRepetition | |
| namespace OAI | |
| /-! # Reusing one round's workspace in the tail-counter repetition | |
| The read-only draws are addressed by the offset counter. A finished round's | |
| workspace is released before advancing to the next draw. | |
| -/ | |
| namespace LowSpaceSubsetSum.DecisionRepetitionCells | |
| variable {Ω : Type*} {k : ℕ} | |
| abbrev Execution := (Bool × ℕ) × ℕ | |
| def loop (step : Ω → Execution) (draws : Fin k → Ω) : | |
| (remaining offset : ℕ) → offset + remaining ≤ k → ℕ → ℕ → Execution | |
| | 0, _, _, work, peak => ((false, work + 1), max peak 8) | |
| | remaining + 1, offset, h, work, peak => | |
| let out := step (draws ⟨offset, by omega⟩) | |
| let used := max peak (out.2 + 8) | |
| if out.1.1 then ((true, work + out.1.2 + 1), used) | |
| else loop step draws remaining (offset + 1) (by omega) (work + out.1.2 + 1) used | |
| def run (step : Ω → Execution) (draws : Fin k → Ω) := | |
| loop step draws k 0 (by omega) 0 8 | |
| theorem loop_value (step : Ω → Execution) (draws : Fin k → Ω) | |
| (remaining offset : ℕ) (h : offset + remaining ≤ k) (work peak : ℕ) : | |
| (loop step draws remaining offset h work peak).1 = | |
| DecisionRepetition.loop (fun x => (step x).1) draws remaining offset h work := by | |
| induction remaining generalizing offset work peak with | |
| | zero => rfl | |
| | succ remaining ih => | |
| rw [loop, DecisionRepetition.loop] | |
| split | |
| · rfl | |
| · exact ih _ _ _ _ | |
| theorem value (step : Ω → Execution) (draws : Fin k → Ω) : | |
| (run step draws).1 = DecisionRepetition.run (fun x => (step x).1) draws := | |
| loop_value step draws k 0 (by omega) 0 8 | |
| theorem loop_cells (step : Ω → Execution) (draws : Fin k → Ω) (L : ℕ) | |
| (hL : ∀ x, (step x).2 ≤ L) (remaining offset : ℕ) | |
| (h : offset + remaining ≤ k) (work peak : ℕ) : | |
| (loop step draws remaining offset h work peak).2 ≤ max peak (L+8) := by | |
| induction remaining generalizing offset work peak with | |
| | zero => | |
| exact max_le_max_left peak (by omega) | |
| | succ remaining ih => | |
| have hx := hL (draws ⟨offset, by omega⟩) | |
| rw [loop] | |
| split | |
| · exact max_le_max_left peak (by omega) | |
| · exact (ih _ _ _ _).trans (max_le (max_le (le_max_left _ _) (by omega)) (le_max_right _ _)) | |
| theorem cells (step : Ω → Execution) (draws : Fin k → Ω) (L : ℕ) | |
| (hL : ∀ x, (step x).2 ≤ L) : (run step draws).2 ≤ L+8 := by | |
| exact (loop_cells step draws L hL k 0 (by omega) 0 8).trans (max_le (by omega) le_rfl) | |
| end LowSpaceSubsetSum.DecisionRepetitionCells | |
| end OAI | |