| import OAI.Computability.UniqueGames.Machines.MachineClone100Certified |
| import OAI.Computability.UniqueGames.Machines.MachineFiniteAlphabet |
| import OAI.Computability.UniqueGames.PCP.Clone100 |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace UniqueGamesTheorem.Explicit.MachineClone100 |
|
|
| open Turing UniqueGamesTheorem.Reduction |
| open UniqueGamesTheorem.Foundations.Complexity |
|
|
| noncomputable section |
|
|
| def indices (t : Outer.Clone100Triples.GoodTriple) : Nat × Nat × Nat := |
| (t.val.1.val, t.val.2.1.val, t.val.2.2.val) |
|
|
| def triples : List (Nat × Nat × Nat) := Outer.Clone100.triples.map indices |
|
|
| theorem triples_length : triples.length = 970200 := by |
| simp [triples, Outer.Clone100.triples_length] |
|
|
| theorem triples_nonempty : triples ≠ [] := by |
| apply List.length_pos_iff.mp |
| rw [triples_length] |
| decide |
|
|
| theorem equation_words {n : Nat} (e : CloneGap.Equation (Fin n)) |
| (t : Outer.Clone100Triples.GoodTriple) : |
| SourceEncoding.equationWords (Outer.Clone100.equation e t) = |
| MachineClone100Table.cloneEquationWords e (indices t) := by |
| simp [SourceEncoding.equationWords, Outer.Clone100.equation, |
| Outer.Clone100.nameEquiv, finProdFinEquiv, |
| MachineClone100Table.cloneEquationWords, indices, Nat.mul_comm] |
|
|
| theorem body_words {n : Nat} (es : List (CloneGap.Equation (Fin n))) : |
| (Outer.Clone100.equations es).flatMap SourceEncoding.equationWords = |
| es.flatMap (fun e => triples.flatMap (MachineClone100Table.cloneEquationWords e)) := by |
| simp only [Outer.Clone100.equations, List.flatMap_assoc, List.flatMap_map, |
| equation_words, triples] |
|
|
| theorem input_words_eq (input : SourceEncoding.Input) : |
| SourceEncoding.inputWords (Outer.Clone100.clonedInput input) = |
| [100 * input.variables, triples.length * input.equations.length] ++ |
| input.equations.flatMap |
| (fun e => triples.flatMap (MachineClone100Table.cloneEquationWords e)) := by |
| simp only [SourceEncoding.inputWords, Outer.Clone100.clonedInput, |
| Outer.Clone100.equations_length, body_words, triples_length] |
| simp [Nat.mul_comm] |
|
|
| theorem input_bits_eq (input : SourceEncoding.Input) : |
| SourceEncoding.inputBits (Outer.Clone100.clonedInput input) = |
| MachineClone100Run.outputBits triples input := by |
| rw [SourceEncoding.inputBits, input_words_eq, encodeWords_append] |
| rfl |
|
|
| private theorem serialized_polynomial (D N n m : Nat) (hn : n ≤ N) (hm : m ≤ N) : |
| n * 100 + m * D + 2 + m * D * (300 * n + 2) ≤ |
| (300 * D) * N * N + (100 + 3 * D) * N + 2 := by |
| have hproduct := Nat.mul_le_mul_left (300 * D) (Nat.mul_le_mul hm hn) |
| have hvariables := Nat.mul_le_mul_left 100 hn |
| have hoccurrences := Nat.mul_le_mul_left (3 * D) hm |
| calc |
| _ = (300 * D) * (m * n) + 100 * n + (3 * D) * m + 2 := by ring |
| _ ≤ (300 * D) * (N * N) + 100 * N + (3 * D) * N + 2 := |
| Nat.add_le_add (Nat.add_le_add (Nat.add_le_add hproduct hvariables) hoccurrences) |
| (Nat.le_refl 2) |
| _ = _ := by ring |
|
|
| theorem size_bound (input : SourceEncoding.Input) : |
| (SourceEncoding.inputBits (Outer.Clone100.clonedInput input)).length ≤ |
| (300 * triples.length) * (SourceEncoding.inputBits input).length * |
| (SourceEncoding.inputBits input).length + |
| (100 + 3 * triples.length) * (SourceEncoding.inputBits input).length + 2 := by |
| rw [triples_length] |
| exact (Outer.Clone100.clonedInput_bits_length_le input).trans |
| (serialized_polynomial 970200 (SourceEncoding.inputBits input).length |
| input.variables input.equations.length |
| (SourceEncoding.inputBits_length_ge_variables input) |
| (SourceEncoding.inputBits_length_ge_equations input)) |
|
|
| def computation : TM2ComputableInPolyTime SourceEncoding.inputBits |
| SourceEncoding.inputBits Outer.Clone100.clonedInput := |
| MachineClone100Certified.certifiedFunction triples triples_nonempty |
| Outer.Clone100.clonedInput input_bits_eq size_bound |
|
|
| theorem finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm := |
| MachineFiniteAlphabet.of_bool _ (fun _ => rfl) |
|
|
| def fullRunInTime (input : SourceEncoding.Input) := computation.outputsFun input |
|
|
| end |
|
|
| end UniqueGamesTheorem.Explicit.MachineClone100 |
|
|
| end |
|
|
| end OAI |
|
|
