| import Mathlib |
| import OAI.Computability.VertexCover.PCP.AlphabetTableTable |
| import OAI.Computability.VertexCover.PCP.AlphabetGraphBounds |
|
|
| namespace OAI |
|
|
| |
|
|
| namespace UniqueGames.Foundations.PCP.AlphabetTableBounds |
|
|
| open AlphabetTable |
|
|
| variable {q : Nat} |
|
|
| theorem gap_transfer_real (hq : 0 < q) (input : GenericGraphTables.Table q) |
| (eps : ℝ) (eps_nonnegative : 0 ≤ eps) |
| (source : ∀ labeling : Fin input.vertices → Fin q, |
| eps * (input.darts : ℝ) ≤ |
| ((GenericGraphTables.semantics input).rejectionCount labeling : ℝ)) |
| (labeling : Fin (Table.build input).vertices → GraphTables.Label) : |
| (eps / 12288) * ((Table.build input).darts : ℝ) ≤ |
| ((GraphTables.semantics (Table.build input)).rejectionCount labeling : ℝ) := by |
| classical |
| let : Nonempty (Fin q) := ⟨⟨0, hq⟩⟩ |
| let lifted := fun vertex => Enumeration.labelEquiv.symm |
| (labeling (Enumeration.vertexEquiv input.vertices input.darts q vertex)) |
| have h := AlphabetGraphBounds.gap_transfer_real |
| (GenericGraphTables.semantics input) eps eps_nonnegative |
| (by simpa only [Fintype.card_fin] using source) lifted |
| rw [Enumeration.card_dart] at h |
| have counts := Table.rejectionCount_build input lifted |
| have restored : (fun v : Fin (Table.build input).vertices => Enumeration.labelEquiv |
| (lifted ((Enumeration.vertexEquiv input.vertices input.darts q).symm v))) = labeling := by |
| funext v |
| dsimp only [lifted] |
| rw [Enumeration.labelEquiv.apply_symm_apply] |
| exact congrArg labeling |
| ((Enumeration.vertexEquiv input.vertices input.darts q).apply_symm_apply v) |
| rw [restored] at counts |
| rw [← counts] at h |
| exact h |
|
|
| theorem build_vertices (input : GenericGraphTables.Table q) : |
| (Table.build input).vertices = input.vertices * 2 ^ q + |
| input.darts * AlphabetGraphBounds.vertexFactor q := by |
| rw [Table.build_vertices] |
| simp only [Enumeration.vertexCount, Enumeration.eventCount, Enumeration.addressCount, |
| Enumeration.localCount, Enumeration.tapeCount, Enumeration.pairTapeCount, |
| Enumeration.fiveTapeCount, AlphabetGraphBounds.vertexFactor, |
| AlphabetGraphBounds.localEventFactor] |
| ring |
|
|
| theorem build_darts (input : GenericGraphTables.Table q) : |
| (Table.build input).darts = input.darts * AlphabetGraphBounds.dartFactor q := by |
| rw [Table.build_darts] |
| simp only [Enumeration.dartCount, Enumeration.eventCount, Enumeration.localCount, |
| Enumeration.tapeCount, Enumeration.pairTapeCount, Enumeration.fiveTapeCount, |
| AlphabetGraphBounds.dartFactor, AlphabetGraphBounds.localEventFactor] |
| ring |
|
|
| theorem build_total (input : GenericGraphTables.Table q) : |
| (Table.build input).vertices + (Table.build input).darts = |
| input.vertices * 2 ^ q + input.darts * |
| (AlphabetGraphBounds.vertexFactor q + AlphabetGraphBounds.dartFactor q) := by |
| rw [build_vertices, build_darts] |
| ring |
|
|
| theorem build_total_le (input : GenericGraphTables.Table q) : |
| (Table.build input).vertices + (Table.build input).darts ≤ |
| AlphabetGraphBounds.sizeFactor q * (input.vertices + input.darts) := by |
| rw [build_total] |
| have hv : 2 ^ q ≤ AlphabetGraphBounds.sizeFactor q := by |
| unfold AlphabetGraphBounds.sizeFactor |
| omega |
| have he : AlphabetGraphBounds.vertexFactor q + AlphabetGraphBounds.dartFactor q ≤ |
| AlphabetGraphBounds.sizeFactor q := by |
| unfold AlphabetGraphBounds.sizeFactor |
| exact Nat.add_le_add_right (Nat.le_add_left _ _) _ |
| calc |
| _ ≤ input.vertices * AlphabetGraphBounds.sizeFactor q + |
| input.darts * AlphabetGraphBounds.sizeFactor q := |
| Nat.add_le_add (Nat.mul_le_mul_left _ hv) (Nat.mul_le_mul_left _ he) |
| _ = _ := by ring |
|
|
| theorem build_vertices_le (input : GenericGraphTables.Table q) : |
| (Table.build input).vertices ≤ |
| AlphabetGraphBounds.sizeFactor q * (input.vertices + input.darts) := |
| (Nat.le_add_right _ _).trans (build_total_le input) |
|
|
| theorem build_darts_le (input : GenericGraphTables.Table q) : |
| (Table.build input).darts ≤ |
| AlphabetGraphBounds.sizeFactor q * (input.vertices + input.darts) := |
| (Nat.le_add_left _ _).trans (build_total_le input) |
|
|
| theorem sizeFactor_positive : 0 < AlphabetGraphBounds.sizeFactor q := |
| AlphabetGraphBounds.sizeFactor_positive q |
|
|
| end UniqueGames.Foundations.PCP.AlphabetTableBounds |
|
|
| end OAI |
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|
