| import Mathlib |
| 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 |
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|
| 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 |
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|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
