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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 | import Mathlib.Algebra.Polynomial.Eval.Defs
import OAI.Computability.UniqueGames.PCP.Clone100
namespace OAI
section
namespace UniqueGamesTheorem.Outer.Clone100Encoding
open UniqueGamesTheorem.Reduction CloneGap
open Clone100
open UniqueGamesTheorem.Foundations.Complexity
noncomputable section
def tripleWords (t : Clone100Triples.GoodTriple) : Nat × Nat × Nat :=
(t.val.1.val, t.val.2.1.val, t.val.2.2.val)
def fixedTriples : List (Nat × Nat × Nat) := Clone100.triples.map tripleWords
theorem fixedTriples_length : fixedTriples.length = 970200 := by
simp [fixedTriples, Clone100.triples_length]
theorem fixedTriples_properties {t : Nat × Nat × Nat} (ht : t ∈ fixedTriples) :
t.1 < 100 ∧ t.2.1 < 100 ∧ t.2.2 < 100 ∧
t.1 ≠ t.2.1 ∧ t.1 ≠ t.2.2 ∧ t.2.1 ≠ t.2.2 := by
obtain ⟨u, _, rfl⟩ := List.mem_map.mp ht
exact ⟨u.val.1.isLt, u.val.2.1.isLt, u.val.2.2.isLt,
fun h => u.property.1 (Fin.ext h),
fun h => u.property.2.1 (Fin.ext h),
fun h => u.property.2.2 (Fin.ext h)⟩
theorem name_value {n : Nat} (v : Fin n) (c : Fin 100) :
(nameEquiv n (v, c)).val = 100 * v.val + c.val := by
simp [nameEquiv, finProdFinEquiv, Nat.mul_comm, Nat.add_comm]
def equationWords {n : Nat} (e : Equation (Fin n)) (t : Nat × Nat × Nat) : List Nat :=
[100 * e.first.val + t.1, 100 * e.second.val + t.2.1,
100 * e.third.val + t.2.2, if e.rhs then 1 else 0]
theorem equation_words {n : Nat} (e : Equation (Fin n)) (t : Clone100Triples.GoodTriple) :
SourceEncoding.equationWords (equation e t) = equationWords e (tripleWords t) := by
cases h : e.rhs <;>
simp [SourceEncoding.equationWords, equation, equationWords, tripleWords, name_value, h]
def bodyWords {n : Nat} (es : List (Equation (Fin n))) : List Nat :=
es.flatMap (fun e => fixedTriples.flatMap (equationWords e))
theorem equations_words {n : Nat} (es : List (Equation (Fin n))) :
(equations es).flatMap SourceEncoding.equationWords = bodyWords es := by
simp only [equations, bodyWords, fixedTriples, List.flatMap_assoc, List.flatMap_map]
apply List.flatMap_congr
intro e _
apply List.flatMap_congr
intro t _
exact equation_words e t
def inputWords (input : SourceEncoding.Input) : List Nat :=
[input.variables * 100, input.equations.length * 970200] ++ bodyWords input.equations
theorem clonedInput_words (input : SourceEncoding.Input) :
SourceEncoding.inputWords (clonedInput input) = inputWords input := by
simp [SourceEncoding.inputWords, clonedInput, inputWords, equations_length, equations_words]
theorem clonedInput_decodes (input : SourceEncoding.Input) :
SourceEncoding.decodeInputWords (inputWords input) = some (clonedInput input) := by
rw [← clonedInput_words]
exact SourceEncoding.decodeInputWords_encoded _
def sizePolynomial : Polynomial Nat :=
Polynomial.C 100 * Polynomial.X + Polynomial.C 970200 * Polynomial.X + Polynomial.C 2 +
(Polynomial.C 970200 * Polynomial.X) * (Polynomial.C 300 * Polynomial.X + Polynomial.C 2)
theorem sizePolynomial_eval (N : Nat) :
sizePolynomial.eval N = 100 * N + 970200 * N + 2 + 970200 * N * (300 * N + 2) := by
simp [sizePolynomial]
theorem clonedInput_bits_polynomial (input : SourceEncoding.Input) :
(SourceEncoding.inputBits (clonedInput input)).length ≤
sizePolynomial.eval (SourceEncoding.inputBits input).length := by
have hn := SourceEncoding.inputBits_length_ge_variables input
have hm := SourceEncoding.inputBits_length_ge_equations input
have ha := Nat.mul_le_mul_right 100 hn
have hb := Nat.mul_le_mul_right 970200 hm
have hc := Nat.mul_le_mul_left 300 hn
have hd := Nat.mul_le_mul hb (Nat.add_le_add_right hc 2)
apply (Clone100.clonedInput_bits_length_le input).trans
rw [sizePolynomial_eval]
nlinarith
end
end UniqueGamesTheorem.Outer.Clone100Encoding
end
end OAI
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