YES(?,PRIMREC)

We are left with following problem, upon which TcT provides the
certificate YES(?,PRIMREC).

Strict Trs:
  { f(0()) -> 0()
  , f(s(0())) -> s(0())
  , f(s(s(x))) -> p(h(g(x)))
  , f(s(s(x))) -> +(p(g(x)), q(g(x)))
  , p(pair(x, y)) -> x
  , h(x) -> pair(+(p(x), q(x)), p(x))
  , g(0()) -> pair(s(0()), s(0()))
  , g(s(x)) -> h(g(x))
  , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
  , +(x, 0()) -> x
  , +(x, s(y)) -> s(+(x, y))
  , q(pair(x, y)) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,PRIMREC)

The input was oriented with the instance of'multiset path order' as
induced by the precedence

 f > s, f > p, f > h, f > g, f > pair, f > +, f > q, 0 > f, 0 > s,
 0 > p, 0 > h, 0 > g, 0 > pair, 0 > +, 0 > q, h > s, h > p,
 h > pair, h > +, h > q, g > s, g > p, g > h, g > pair, g > +,
 g > q, pair > s, pair > p, pair > q, + > s, + > p, + > pair, + > q,
 p ~ q .

Hurray, we answered YES(?,PRIMREC)