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import OAI.Computability.VertexCover.PCP.GenericGraphTables
namespace OAI
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namespace UniqueGames.Foundations.PCP.GraphTableComplexity
open UniqueGames.Foundations.Complexity
open Polynomial
noncomputable def encodingPolynomial (q : Nat) : Polynomial Nat :=
X ^ 2 + C (2 * (q * q) + 1) * X + C 2
theorem encodingPolynomial_eval (q n : Nat) :
(encodingPolynomial q).eval n = n ^ 2 + (2 * (q * q) + 1) * n + 2 := by
simp [encodingPolynomial]
theorem encodingPolynomial_mono (q : Nat) {n m : Nat} (hnm : n ≤ m) :
(encodingPolynomial q).eval n ≤ (encodingPolynomial q).eval m := by
rw [encodingPolynomial_eval, encodingPolynomial_eval]
exact Nat.add_le_add_right
(Nat.add_le_add (Nat.pow_le_pow_left hnm 2) (Nat.mul_le_mul_left _ hnm)) 2
theorem rowBound_le_polynomial (q vertices darts : Nat) :
vertices + darts + 2 + darts * (vertices + darts + 2 * (q * q)) ≤
(encodingPolynomial q).eval (vertices + darts) := by
have hd : darts ≤ vertices + darts := Nat.le_add_left darts vertices
calc
_ ≤ vertices + darts + 2 +
(vertices + darts) * (vertices + darts + 2 * (q * q)) :=
Nat.add_le_add_left (Nat.mul_le_mul_right _ hd) _
_ = _ := by rw [encodingPolynomial_eval]; ring
theorem generic_size_add_two_le_bits {q : Nat} (table : GenericGraphTables.Table q) :
table.vertices + table.darts + 2 ≤ (GenericGraphTables.tableBits table).length := by
simp only [GenericGraphTables.tableBits, GenericGraphTables.tableWords,
encodeWords_append, List.length_append, encodeWords, encodeWord_length, List.length_nil]
omega
theorem generic_bits_le_polynomial {q : Nat} (table : GenericGraphTables.Table q) :
(GenericGraphTables.tableBits table).length ≤
(encodingPolynomial q).eval (table.vertices + table.darts) :=
(GenericGraphTables.tableBits_length_le table).trans
(rowBound_le_polynomial q table.vertices table.darts)
theorem generic_bits_le_of_size_le {q N : Nat} (table : GenericGraphTables.Table q)
(hsize : table.vertices + table.darts ≤ N) :
(GenericGraphTables.tableBits table).length ≤ (encodingPolynomial q).eval N :=
(generic_bits_le_polynomial table).trans (encodingPolynomial_mono q hsize)
theorem size_add_two_le_bits (table : GraphTables.Table) :
table.vertices + table.darts + 2 ≤ (GraphTables.tableBits table).length := by
simp only [GraphTables.tableBits, GraphTables.tableWords,
encodeWords_append, List.length_append, encodeWords, encodeWord_length, List.length_nil]
omega
theorem bits_le_polynomial (table : GraphTables.Table) :
(GraphTables.tableBits table).length ≤
(encodingPolynomial 64).eval (table.vertices + table.darts) :=
(GraphTables.tableBits_length_le table).trans
(rowBound_le_polynomial 64 table.vertices table.darts)
theorem bits_le_of_size_le {N : Nat} (table : GraphTables.Table)
(hsize : table.vertices + table.darts ≤ N) :
(GraphTables.tableBits table).length ≤ (encodingPolynomial 64).eval N :=
(bits_le_polynomial table).trans (encodingPolynomial_mono 64 hsize)
end UniqueGames.Foundations.PCP.GraphTableComplexity
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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