openAI-math-dataset / lean /OAI /Computability /UniqueGames /Machines /MachineClone100Certified.lean
| import OAI.Computability.UniqueGames.Machines.MachineClone100Run | |
| namespace OAI | |
| section | |
| namespace UniqueGamesTheorem.Explicit.MachineClone100Certified | |
| open UniqueGamesTheorem.Reduction | |
| open Turing | |
| open UniqueGamesTheorem.Foundations.Complexity | |
| noncomputable def timePolynomial (copies : Nat) : Polynomial Nat := | |
| Polynomial.C (300 * copies) * Polynomial.X * Polynomial.X + | |
| Polynomial.C (120 + 18 * copies) * Polynomial.X + Polynomial.C 23 | |
| theorem timePolynomial_eval (copies length : Nat) : | |
| (timePolynomial copies).eval length = | |
| (300 * copies) * length * length + (120 + 18 * copies) * length + 23 := by | |
| simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul, | |
| Polynomial.eval_C, Polynomial.eval_X] | |
| private theorem numericalBudget (D N n m B out : Nat) | |
| (hn : n ≤ N) (hm : m ≤ N) (hB : B ≤ N) | |
| (ho : out ≤ (300 * D) * N * N + (100 + 3 * D) * N + 2) : | |
| (out + 2) + (((2 + 3 * D) * B + (9 + 12 * D) * m + 1) + | |
| (5 * n + 4 * m + 18)) ≤ (300 * D) * N * N + (120 + 18 * D) * N + 23 := by | |
| have hn' := Nat.mul_le_mul_left 5 hn | |
| have hm' := Nat.mul_le_mul_left (13 + 12 * D) hm | |
| have hb' := Nat.mul_le_mul_left (2 + 3 * D) hB | |
| calc | |
| _ = 5 * n + (13 + 12 * D) * m + (2 + 3 * D) * B + out + 21 := by ring | |
| _ ≤ 5 * N + (13 + 12 * D) * N + (2 + 3 * D) * N + | |
| ((300 * D) * N * N + (100 + 3 * D) * N + 2) + 21 := by omega | |
| _ = _ := by ring | |
| noncomputable def certifiedFunction (triples : List (CloneGap.Index × CloneGap.Index × CloneGap.Index)) | |
| (nonempty : triples ≠ []) (f : SourceEncoding.Input → SourceEncoding.Input) | |
| (output_eq : ∀ input, SourceEncoding.inputBits (f input) = | |
| MachineClone100Run.outputBits triples input) | |
| (size_bound : ∀ input, | |
| (SourceEncoding.inputBits (f input)).length ≤ | |
| (300 * triples.length) * (SourceEncoding.inputBits input).length * | |
| (SourceEncoding.inputBits input).length + | |
| (100 + 3 * triples.length) * (SourceEncoding.inputBits input).length + 2) : | |
| TM2ComputableInPolyTime SourceEncoding.inputBits SourceEncoding.inputBits f where | |
| tm := MachineClone100Run.tm triples nonempty | |
| inputAlphabet := Equiv.refl Bool | |
| outputAlphabet := Equiv.refl Bool | |
| time := timePolynomial triples.length | |
| outputsFun input := by | |
| change TM2OutputsInTime (MachineClone100Run.tm triples nonempty) | |
| ((SourceEncoding.inputBits input).map id) | |
| (some ((SourceEncoding.inputBits (f input)).map id)) | |
| ((timePolynomial triples.length).eval (SourceEncoding.inputBits input).length) | |
| rw [@List.map_id ((MachineClone100Run.tm triples nonempty).Γ | |
| (MachineClone100Run.tm triples nonempty).k₀) (SourceEncoding.inputBits input), | |
| @List.map_id ((MachineClone100Run.tm triples nonempty).Γ | |
| (MachineClone100Run.tm triples nonempty).k₁) (SourceEncoding.inputBits (f input)), | |
| output_eq input, timePolynomial_eval] | |
| let actual := MachineClone100Run.runInTime triples nonempty input | |
| refine { toEvalsTo := actual.toEvalsTo, steps_le_m := actual.steps_le_m.trans ?_ } | |
| apply numericalBudget | |
| · exact SourceEncoding.inputBits_length_ge_variables input | |
| · exact SourceEncoding.inputBits_length_ge_equations input | |
| · have h := SourceEncoding.inputBits_length input | |
| change (SourceEncoding.inputBits input).length = | |
| input.variables + input.equations.length + 2 + | |
| (MachineClone100Run.sourceBody input).length at h | |
| omega | |
| · rw [← output_eq input] | |
| exact size_bound input | |
| end UniqueGamesTheorem.Explicit.MachineClone100Certified | |
| end | |
| end OAI | |