openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Counting /GuessArrayExecution.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.CanonicalGuesses | |
| namespace OAI | |
| /-! # Three bounded interval draws stored as a small integer array -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.GuessArrayExecution | |
| open scoped BigOperators | |
| def coupling (n : ℕ) : SamplerCoupling (RepresentationSetup.Guesses n) := | |
| SamplerCoupling.pi 3 (fun _ => ↥(RepresentationSetup.guessRange n)) | |
| (fun _ => CanonicalGuesses.coupling n) | |
| def execution (n : ℕ) : SamplerExecution.Execution (coupling n) := | |
| SamplerExecution.pi 3 _ _ (fun _ => CanonicalGuesses.execution n) | |
| theorem execution_bounded (n : ℕ) : (execution n).Bounded (3 * n ^ 3) 0 := by | |
| have h := SamplerExecution.pi_bounded 3 (fun _ => ↥(RepresentationSetup.guessRange n)) | |
| (fun _ => CanonicalGuesses.coupling n) (fun _ => CanonicalGuesses.execution n) | |
| (fun _ => n ^ 3) (fun _ => 0) (fun _ => CanonicalGuesses.execution_bounded n) | |
| simpa only [execution, coupling, Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul, | |
| mul_zero] using h | |
| def run (n : ℕ) (s : (coupling n).Seed) : Option (Vector ℕ 3) × ℕ := | |
| let out := (execution n).run s | |
| (out.value.map (fun g => Vector.ofFn (fun i => (g i).val)), out.reads + out.tests + 7) | |
| theorem value (n : ℕ) (s : (coupling n).Seed) : | |
| (run n s).1 = ((coupling n).bounded s).map (fun g => Vector.ofFn (fun i => (g i).val)) := by | |
| simp only [run, (execution n).value_eq] | |
| theorem work (n : ℕ) (s : (coupling n).Seed) : (run n s).2 ≤ 3 * n ^ 3 + 7 := by | |
| have h := execution_bounded n s | |
| change (run n s).2 ≤ _ | |
| change ((execution n).run s).reads + ((execution n).run s).tests + 7 ≤ _ | |
| omega | |
| end LowSpaceSubsetSum.GuessArrayExecution | |
| end | |
| end OAI | |