openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordSeedModulusExecution.lean
| import Mathlib.Data.Nat.Log | |
| namespace OAI | |
| /-! # Computing a rejection range width and its cached modulus | |
| A doubling cursor starts at one and stops at the first power of two at | |
| least the requested range size. The width and modulus are computed together. | |
| -/ | |
| namespace LowSpaceSubsetSum.WordSeedModulusExecution | |
| structure Result where | |
| width : ℕ | |
| modulus : ℕ | |
| work : ℕ | |
| def loop (D : ℕ) : ℕ → ℕ → ℕ → ℕ → Result | |
| | 0, width, modulus, work => ⟨width,modulus,work+1⟩ | |
| | fuel+1, width, modulus, work => | |
| if modulus < D then loop D fuel (width+1) (2*modulus) (work+5) | |
| else ⟨width,modulus,work+1⟩ | |
| theorem loop_value (D fuel width modulus work : ℕ) | |
| (hm : modulus = 2^width) (hw : width ≤ Nat.clog 2 D) | |
| (hf : Nat.clog 2 D ≤ width+fuel) : | |
| loop D fuel width modulus work = | |
| ⟨Nat.clog 2 D,2^Nat.clog 2 D,work+5*(Nat.clog 2 D-width)+1⟩ := by | |
| induction fuel generalizing width modulus work with | |
| | zero => | |
| have he : width = Nat.clog 2 D := by omega | |
| subst width | |
| subst modulus | |
| simp only [loop,Nat.sub_self,Nat.mul_zero,Nat.add_zero] | |
| | succ fuel ih => | |
| change (if modulus < D then loop D fuel (width+1) (2*modulus) (work+5) | |
| else ⟨width,modulus,work+1⟩) = _ | |
| split | |
| next h => | |
| have hlt : width < Nat.clog 2 D := by | |
| exact (Nat.lt_clog_iff_pow_lt (by decide : 1 < 2)).mpr (hm ▸ h) | |
| have hm' : 2*modulus = 2^(width+1) := by rw [hm,Nat.pow_succ]; omega | |
| rw [ih _ _ _ hm' (by omega) (by omega)] | |
| congr 1 | |
| omega | |
| next h => | |
| have hge : Nat.clog 2 D ≤ width := by | |
| apply Nat.clog_le_of_le_pow | |
| omega | |
| have he : width = Nat.clog 2 D := by omega | |
| subst width | |
| subst modulus | |
| simp only [Nat.sub_self,Nat.mul_zero,Nat.add_zero] | |
| def run (D : ℕ) : Result := loop D D 0 1 0 | |
| theorem run_value (D : ℕ) : run D = ⟨Nat.clog 2 D,2^Nat.clog 2 D,5*Nat.clog 2 D+1⟩ := by | |
| have hf : Nat.clog 2 D ≤ D := Nat.clog_le_of_le_pow (Nat.lt_two_pow_self.le) | |
| simpa only [run,Nat.pow_zero,Nat.zero_add,Nat.sub_zero] using | |
| loop_value D D 0 1 0 (by simp) (Nat.zero_le _) (by omega) | |
| theorem run_width (D : ℕ) : (run D).width = Nat.clog 2 D := by rw [run_value] | |
| theorem run_modulus (D : ℕ) : (run D).modulus = 2^Nat.clog 2 D := by rw [run_value] | |
| theorem run_work (D : ℕ) : (run D).work = 5*Nat.clog 2 D+1 := by rw [run_value] | |
| theorem run_modulus_fits (D w : ℕ) (hw : Nat.clog 2 D < w) : | |
| (run D).modulus < 2^w := by | |
| rw [run_modulus] | |
| exact Nat.pow_lt_pow_right (by decide : 1 < 2) hw | |
| end LowSpaceSubsetSum.WordSeedModulusExecution | |
| end OAI | |