openAI-math-dataset / lean /OAI /Combinatorics /IndependentSets /Encoding /GraphTableComplexity.lean
| import Mathlib | |
| import OAI.Combinatorics.IndependentSets.PCP.GraphTables | |
| import OAI.Combinatorics.IndependentSets.PCP.GenericGraphTables | |
| namespace OAI | |
| namespace IndependentSetsGames.Foundations.PCP.GraphTableComplexity | |
| open IndependentSetsGames.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 IndependentSetsGames.Foundations.PCP.GraphTableComplexity | |
| end OAI | |