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2f887db | 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 | import Std
import OAI.Computability.PerfectCompleteness.Model
namespace OAI
section
namespace PerfectCompleteness
theorem countSatisfied_le_length {l r q : Nat} (a : Fin l → Fin (2 * q))
(b : Fin r → Fin q) (es : List (Edge l r q)) :
countSatisfied a b es ≤ es.length := by
induction es with
| nil => simp [countSatisfied]
| cons e es ih =>
simp only [countSatisfied, List.length_cons]
split <;> omega
theorem countSatisfied_eq_length_iff {l r q : Nat} (a : Fin l → Fin (2 * q))
(b : Fin r → Fin q) (es : List (Edge l r q)) :
countSatisfied a b es = es.length ↔ ∀ e ∈ es, e.satisfied a b = true := by
induction es with
| nil => simp [countSatisfied]
| cons e es ih =>
have bound := countSatisfied_le_length a b es
cases h : e.satisfied a b <;> simp_all [countSatisfied, Nat.add_comm]
omega
theorem Instance.edgeCount_positive {q : Nat} (G : Instance q) :
0 < G.edges.length := by
cases h : G.edges with
| nil => exact False.elim (G.nonempty h)
| cons e es => simp
def PerfectlyComplete {q : Nat} (G : Instance q) : Prop :=
∃ s : Labeling G, ∀ e ∈ G.edges, e.satisfied s.1 s.2 = true
structure RationalThreshold where
numerator : Nat
denominator : Nat
positive : 0 < numerator
belowOne : numerator < denominator
def SoundAt {q : Nat} (d : RationalThreshold) (G : Instance q) : Prop :=
∀ s : Labeling G,
d.denominator * countSatisfied s.1 s.2 G.edges ≤ d.numerator * G.edges.length
end PerfectCompleteness
end
end OAI
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