| import Mathlib |
| import OAI.Computability.VertexCover.PCP.AlphabetGraph |
|
|
| namespace OAI |
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| |
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|
| namespace UniqueGames.Foundations.PCP.AlphabetGraphBounds |
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|
| open scoped BigOperators |
|
|
| abbrev OutputVertex (V E A : Type*) := |
| QueryIncidence.Vertex (AlphabetGraph.Event E A) (AlphabetGraph.Address V E A) |
|
|
| abbrev OutputDart (E A : Type*) := QueryIncidence.Dart (AlphabetGraph.Event E A) 6 |
|
|
| def localEventFactor (q : Nat) : Nat := |
| 4 * (2 * 2 ^ q) * (2 ^ (q * q)) ^ 5 |
|
|
| def vertexFactor (q : Nat) : Nat := localEventFactor q + 2 ^ (q * q) |
|
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| def dartFactor (q : Nat) : Nat := 12 * localEventFactor q |
|
|
| def sizeFactor (q : Nat) : Nat := 2 ^ q + vertexFactor q + dartFactor q |
|
|
| theorem sizeFactor_positive (q : Nat) : 0 < sizeFactor q := by |
| have hp : (0 : Nat) < 2 ^ q := pow_pos (by decide) q |
| unfold sizeFactor |
| omega |
|
|
| variable {V E A : Type*} [Fintype A] [DecidableEq A] [Nonempty A] |
|
|
| theorem gap_transfer_real [Fintype E] [DecidableEq V] [DecidableEq E] |
| (G : ConstraintGraph V E A) (eps : ℝ) (_eps_nonnegative : 0 ≤ eps) |
| (source : ∀ labeling : V → A, |
| eps * (Fintype.card E : ℝ) ≤ (G.rejectionCount labeling : ℝ)) |
| (labeling : OutputVertex V E A → QueryIncidence.Label 6) : |
| (eps / 12288) * (Fintype.card (OutputDart E A) : ℝ) ≤ |
| ((AlphabetGraph.graph G).rejectionCount labeling : ℝ) := by |
| let assignment := QueryIncidence.decodedAssignment (by decide : 0 < 6) labeling |
| have htest : |
| (Fintype.card (AlphabetGraph.LocalEvent A) : ℝ) * |
| (G.rejectionCount (AlphabetGraph.decodedLabeling assignment) : ℝ) ≤ |
| 2048 * (QueryIncidence.verifierRejectionCount (AlphabetGraph.verifier G) assignment : ℝ) := by |
| exact_mod_cast AlphabetGraph.rejection_count_bound G assignment |
| have hinc : |
| 2 * (QueryIncidence.verifierRejectionCount (AlphabetGraph.verifier G) assignment : ℝ) ≤ |
| ((AlphabetGraph.graph G).rejectionCount labeling : ℝ) := by |
| exact_mod_cast QueryIncidence.rejection_count_bound |
| (AlphabetGraph.verifier G) (by decide : 0 < 6) labeling |
| have hscaled : |
| eps * (Fintype.card E : ℝ) * (Fintype.card (AlphabetGraph.LocalEvent A) : ℝ) ≤ |
| 2048 * (QueryIncidence.verifierRejectionCount (AlphabetGraph.verifier G) assignment : ℝ) := by |
| calc |
| _ = (Fintype.card (AlphabetGraph.LocalEvent A) : ℝ) * |
| (eps * (Fintype.card E : ℝ)) := by ring |
| _ ≤ (Fintype.card (AlphabetGraph.LocalEvent A) : ℝ) * |
| (G.rejectionCount (AlphabetGraph.decodedLabeling assignment) : ℝ) := |
| mul_le_mul_of_nonneg_left (source (AlphabetGraph.decodedLabeling assignment)) |
| (Nat.cast_nonneg _) |
| _ ≤ _ := htest |
| change (eps / 12288) * |
| (Fintype.card (QueryIncidence.Dart (AlphabetGraph.Event E A) 6) : ℝ) ≤ _ |
| rw [QueryIncidence.card_dart, AlphabetGraph.card_event] |
| push_cast |
| nlinarith |
|
|
| variable [Fintype V] [Fintype E] |
|
|
| omit [Nonempty A] in |
| theorem card_output_vertex : |
| Fintype.card (OutputVertex V E A) = |
| Fintype.card V * 2 ^ Fintype.card A + |
| Fintype.card E * vertexFactor (Fintype.card A) := by |
| rw [QueryIncidence.card_vertex, AlphabetGraph.card_event, |
| AlphabetGraph.card_address, AlphabetGraph.card_localEvent] |
| unfold vertexFactor localEventFactor |
| ring |
|
|
| omit [Nonempty A] in |
| theorem card_output_dart : |
| Fintype.card (OutputDart E A) = Fintype.card E * dartFactor (Fintype.card A) := by |
| rw [QueryIncidence.card_dart, AlphabetGraph.card_event, AlphabetGraph.card_localEvent] |
| unfold dartFactor localEventFactor |
| ring |
|
|
| omit [Nonempty A] in |
| theorem card_output_total : |
| Fintype.card (OutputVertex V E A) + Fintype.card (OutputDart E A) = |
| Fintype.card V * 2 ^ Fintype.card A + |
| Fintype.card E * (vertexFactor (Fintype.card A) + dartFactor (Fintype.card A)) := by |
| rw [card_output_vertex, card_output_dart] |
| ring |
|
|
| omit [Nonempty A] in |
|
|
| theorem card_output_total_le : |
| Fintype.card (OutputVertex V E A) + Fintype.card (OutputDart E A) ≤ |
| sizeFactor (Fintype.card A) * (Fintype.card V + Fintype.card E) := by |
| rw [card_output_total] |
| have hv : 2 ^ Fintype.card A ≤ sizeFactor (Fintype.card A) := by |
| unfold sizeFactor |
| omega |
| have he : vertexFactor (Fintype.card A) + dartFactor (Fintype.card A) ≤ |
| sizeFactor (Fintype.card A) := by |
| unfold sizeFactor |
| exact Nat.add_le_add_right (Nat.le_add_left _ _) _ |
| calc |
| _ ≤ Fintype.card V * sizeFactor (Fintype.card A) + |
| Fintype.card E * sizeFactor (Fintype.card A) := |
| Nat.add_le_add (Nat.mul_le_mul_left _ hv) (Nat.mul_le_mul_left _ he) |
| _ = _ := by ring |
|
|
| omit [Nonempty A] in |
| theorem card_output_vertex_le : |
| Fintype.card (OutputVertex V E A) ≤ |
| sizeFactor (Fintype.card A) * (Fintype.card V + Fintype.card E) := |
| (Nat.le_add_right _ _).trans card_output_total_le |
|
|
| omit [Nonempty A] in |
| theorem card_output_dart_le : |
| Fintype.card (OutputDart E A) ≤ |
| sizeFactor (Fintype.card A) * (Fintype.card V + Fintype.card E) := |
| (Nat.le_add_left _ _).trans card_output_total_le |
|
|
| end UniqueGames.Foundations.PCP.AlphabetGraphBounds |
|
|
| end OAI |
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|
