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import OAI.Computability.VertexCover.PCP.ExpanderFamily
import OAI.Computability.VertexCover.PCP.GraphGap
import OAI.Computability.VertexCover.PCP.InitialGraph
namespace OAI
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noncomputable section
namespace UniqueGames.Foundations.PCP.FiniteGraph
structure Bundle (A : Type) where
Vertex : Type
Dart : Type
vertexFintype : Fintype Vertex
dartFintype : Fintype Dart
vertexDecidableEq : DecidableEq Vertex
dartDecidableEq : DecidableEq Dart
dartNonempty : Nonempty Dart
graph : ConstraintGraph Vertex Dart A
namespace Bundle
variable {A : Type}
instance instFintypeVertex (G : Bundle A) : Fintype G.Vertex := G.vertexFintype
instance instFintypeDart (G : Bundle A) : Fintype G.Dart := G.dartFintype
instance instDecidableEqVertex (G : Bundle A) : DecidableEq G.Vertex := G.vertexDecidableEq
instance instDecidableEqDart (G : Bundle A) : DecidableEq G.Dart := G.dartDecidableEq
instance instNonemptyDart (G : Bundle A) : Nonempty G.Dart := G.dartNonempty
instance instNonemptyVertex (G : Bundle A) : Nonempty G.Vertex := by
obtain ⟨d⟩ := G.dartNonempty
exact ⟨G.graph.tail d⟩
def ofGraph {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E]
(G : ConstraintGraph V E A) : Bundle A where
Vertex := V
Dart := E
vertexFintype := inferInstance
dartFintype := inferInstance
vertexDecidableEq := inferInstance
dartDecidableEq := inferInstance
dartNonempty := inferInstance
graph := G
def size (G : Bundle A) : Nat := Fintype.card G.Vertex + Fintype.card G.Dart
def Satisfiable (G : Bundle A) : Prop := G.graph.Satisfiable
def rejectionCount (G : Bundle A) (labeling : G.Vertex → A) : Nat :=
G.graph.rejectionCount labeling
theorem size_positive (G : Bundle A) : 0 < G.size :=
lt_of_lt_of_le (Fintype.card_pos : 0 < Fintype.card G.Dart) (Nat.le_add_left _ _)
theorem dartCard_le_size (G : Bundle A) : Fintype.card G.Dart ≤ G.size :=
Nat.le_add_left _ _
@[simp] theorem ofGraph_graph {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
(ofGraph G).graph = G := rfl
@[simp] theorem ofGraph_size {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
(ofGraph G).size = Fintype.card V + Fintype.card E := rfl
@[simp] theorem ofGraph_satisfiable {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
(ofGraph G).Satisfiable ↔ G.Satisfiable := Iff.rfl
@[simp] theorem ofGraph_rejectionCount {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E]
(G : ConstraintGraph V E A) (labeling : V → A) :
(ofGraph G).rejectionCount labeling = G.rejectionCount labeling := rfl
section Gap
variable [Fintype A] [Nonempty A]
def minimumRejections (G : Bundle A) : Nat := G.graph.minimumRejections
def gap (G : Bundle A) : ℝ := G.graph.gap
theorem exists_minimizer (G : Bundle A) :
∃ labeling : G.Vertex → A, G.rejectionCount labeling = G.minimumRejections :=
G.graph.exists_minimizer
theorem gap_nonnegative (G : Bundle A) : 0 ≤ G.gap := G.graph.gap_nonnegative
theorem gap_le_one (G : Bundle A) : G.gap ≤ 1 := G.graph.gap_le_one
theorem gap_eq_zero_iff (G : Bundle A) : G.gap = 0 ↔ G.Satisfiable :=
G.graph.gap_eq_zero_iff
theorem le_gap_iff (G : Bundle A) (ε : ℝ) :
ε ≤ G.gap ↔ ∀ labeling : G.Vertex → A,
ε * Fintype.card G.Dart ≤ (G.rejectionCount labeling : ℝ) :=
G.graph.le_gap_iff ε
theorem inverse_card_le_gap_of_unsatisfiable (G : Bundle A) (unsat : ¬ G.Satisfiable) :
1 / (Fintype.card G.Dart : ℝ) ≤ G.gap :=
G.graph.inverse_card_le_gap_of_unsatisfiable unsat
theorem one_le_dartCard_mul_gap (G : Bundle A) (unsat : ¬ G.Satisfiable) :
1 ≤ (Fintype.card G.Dart : ℝ) * G.gap := by
have he : (0 : ℝ) < Fintype.card G.Dart := by exact_mod_cast Fintype.card_pos
have h := (div_le_iff₀ he).mp (G.inverse_card_le_gap_of_unsatisfiable unsat)
simpa only [mul_comm] using h
theorem one_le_size_mul_gap (G : Bundle A) (unsat : ¬ G.Satisfiable) :
1 ≤ (G.size : ℝ) * G.gap := by
have hsize : (Fintype.card G.Dart : ℝ) ≤ (G.size : ℝ) := by
exact_mod_cast G.dartCard_le_size
exact (G.one_le_dartCard_mul_gap unsat).trans
(mul_le_mul_of_nonneg_right hsize G.gap_nonnegative)
@[simp] theorem ofGraph_minimumRejections {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
(ofGraph G).minimumRejections = G.minimumRejections := rfl
@[simp] theorem ofGraph_gap {V E : Type} [Fintype V] [Fintype E]
[DecidableEq V] [DecidableEq E] [Nonempty E] (G : ConstraintGraph V E A) :
(ofGraph G).gap = G.gap := rfl
end Gap
end Bundle
end UniqueGames.Foundations.PCP.FiniteGraph
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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