Approximate by thinning: Deriving fully polynomial-time approximation schemes

We check that, letting Q := ( ), S and satisfy condition 3. Note that S† = [〈nil , const 1〉, 〈S′ ◦ (id × fst), suc ◦ snd ◦ snd〉]. Base case: 〈nil , const 1〉 ⊆ d e+ ◦ 〈nil , const 1〉 ≡ ([ ], 1) d e+ ([ ], 1) ≡ { definition of d e+ } ([ ], 1) δ 0 ([ ], 1) ≡ { definition of δ } [ ] ≤v [ ] ∧ [ ] ≥w [ ] ≡ true. For the non-empty case, let S′ = scons ∪ snd , we need that for a :: Item, x1, y1, y2 :: [Item], n :: N, and δ :: R ≥ 1: (x1, n+ 1) S ′† (a, (y1, n)) ∧ (y1, n) d e? (y2, n) ⇒ (∃x2 |: (x1, n+ 1) d e+ (x2, n+ 1) ∧ (x2, n+ 1) S ′† (a, (y2, n))).