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import OAI.Computability.VertexCover.PCP.AlphabetGraph
namespace OAI
namespace UniqueGames.Foundations.PCP.AlphabetGraphBounds
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)
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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