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760f90e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | 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
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