File size: 2,822 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import Mathlib
import OAI.Computability.VertexCover.Encoding.BinaryFormula
namespace OAI
namespace UniqueGames.Foundations.PCP
structure ConstraintGraph (V E A : Type*) where
reverse : E ≃ E
reverse_involutive : Function.Involutive reverse
tail : E → V
accepts : E → A → A → Bool
reverse_accepts : ∀ e a b, accepts (reverse e) b a = accepts e a b
namespace ConstraintGraph
variable {V E A : Type*}
def head (G : ConstraintGraph V E A) (e : E) : V := G.tail (G.reverse e)
@[simp] theorem head_reverse (G : ConstraintGraph V E A) (e : E) :
G.head (G.reverse e) = G.tail e := by
simp only [head, G.reverse_involutive e]
def edgeSatisfied (G : ConstraintGraph V E A) (labeling : V → A) (e : E) : Bool :=
G.accepts e (labeling (G.tail e)) (labeling (G.head e))
@[simp] theorem edgeSatisfied_reverse (G : ConstraintGraph V E A)
(labeling : V → A) (e : E) :
G.edgeSatisfied labeling (G.reverse e) = G.edgeSatisfied labeling e := by
simpa only [edgeSatisfied, head, G.reverse_involutive e] using
G.reverse_accepts e (labeling (G.tail e)) (labeling (G.tail (G.reverse e)))
def Satisfiable (G : ConstraintGraph V E A) : Prop :=
∃ labeling : V → A, ∀ e, G.edgeSatisfied labeling e = true
def rejectedDarts [Fintype E] (G : ConstraintGraph V E A) (labeling : V → A) :
Finset E := Finset.univ.filter (fun e => G.edgeSatisfied labeling e = false)
def rejectionCount [Fintype E] (G : ConstraintGraph V E A) (labeling : V → A) : Nat :=
(G.rejectedDarts labeling).card
theorem mem_rejectedDarts [Fintype E] (G : ConstraintGraph V E A)
(labeling : V → A) (e : E) :
e ∈ G.rejectedDarts labeling ↔ G.edgeSatisfied labeling e = false := by
simp [rejectedDarts]
theorem rejectionCount_le [Fintype E] (G : ConstraintGraph V E A)
(labeling : V → A) : G.rejectionCount labeling ≤ Fintype.card E := by
exact Finset.card_le_card (Finset.filter_subset _ _)
theorem rejected_exists_of_not_satisfiable (G : ConstraintGraph V E A)
(unsat : ¬ G.Satisfiable) (labeling : V → A) :
∃ e, G.edgeSatisfied labeling e = false := by
classical
by_contra none
apply unsat
refine ⟨labeling, fun e => ?_⟩
have notFalse : G.edgeSatisfied labeling e ≠ false := by
intro h
exact none ⟨e, h⟩
cases h : G.edgeSatisfied labeling e <;> simp_all
theorem rejectionCount_positive [Fintype E] (G : ConstraintGraph V E A)
(unsat : ¬ G.Satisfiable) (labeling : V → A) :
1 ≤ G.rejectionCount labeling := by
obtain ⟨e, he⟩ := G.rejected_exists_of_not_satisfiable unsat labeling
exact Finset.one_le_card.mpr ⟨e, (G.mem_rejectedDarts labeling e).mpr he⟩
end ConstraintGraph
end UniqueGames.Foundations.PCP
end OAI
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