openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordSeedPowerExecution.lean
| import Mathlib.Data.Nat.Log | |
| namespace OAI | |
| /-! # Constant-state power-of-two initialization | |
| The same executed doubling cursor supplies both prime candidate moduli | |
| and trial-division limits. | |
| -/ | |
| namespace LowSpaceSubsetSum.WordSeedPowerExecution | |
| def loop : ℕ → ℕ → ℕ → ℕ × ℕ | |
| | 0, value, work => (value,work+1) | |
| | fuel+1, value, work => loop fuel (2*value) (work+3) | |
| theorem loop_value (fuel value work : ℕ) : | |
| loop fuel value work = (value*2^fuel,work+3*fuel+1) := by | |
| induction fuel generalizing value work with | |
| | zero => simp only [loop,Nat.pow_zero,Nat.mul_one,Nat.mul_zero,Nat.add_zero] | |
| | succ fuel ih => | |
| change loop fuel (2*value) (work+3) = _ | |
| rw [ih,Nat.pow_succ] | |
| apply Prod.ext | |
| · simp only [Nat.mul_assoc,Nat.mul_comm] | |
| · omega | |
| def run (width : ℕ) := loop width 1 0 | |
| theorem run_value (width : ℕ) : (run width).1 = 2^width := by | |
| simp only [run,loop_value,Nat.one_mul] | |
| theorem run_work (width : ℕ) : (run width).2 = 3*width+1 := by | |
| simp only [run,loop_value,Nat.zero_add] | |
| end LowSpaceSubsetSum.WordSeedPowerExecution | |
| end OAI | |