YES(?,O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict Trs:
  { f(m, 0()) -> g(m, 0())
  , f(0(), n) -> g(0(), n)
  , f(s(m), s(n)) -> h(m, n, f(m, p(m, n)), f(s(m), n))
  , g(n, m) -> bot()
  , h(k, l, m, n) -> bot()
  , p(m, n) -> bot() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

We add following dependency tuples:

Strict DPs:
  { f^#(m, 0()) -> c_1(g^#(m, 0()))
  , f^#(0(), n) -> c_2(g^#(0(), n))
  , f^#(s(m), s(n)) ->
    c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
        f^#(m, p(m, n)),
        p^#(m, n),
        f^#(s(m), n))
  , g^#(n, m) -> c_4()
  , h^#(k, l, m, n) -> c_5()
  , p^#(m, n) -> c_6() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(m, 0()) -> c_1(g^#(m, 0()))
  , f^#(0(), n) -> c_2(g^#(0(), n))
  , f^#(s(m), s(n)) ->
    c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
        f^#(m, p(m, n)),
        p^#(m, n),
        f^#(s(m), n))
  , g^#(n, m) -> c_4()
  , h^#(k, l, m, n) -> c_5()
  , p^#(m, n) -> c_6() }
Weak Trs:
  { f(m, 0()) -> g(m, 0())
  , f(0(), n) -> g(0(), n)
  , f(s(m), s(n)) -> h(m, n, f(m, p(m, n)), f(s(m), n))
  , g(n, m) -> bot()
  , h(k, l, m, n) -> bot()
  , p(m, n) -> bot() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

We estimate the number of application of {4,5,6} by applications of
Pre({4,5,6}) = {1,2,3}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(m, 0()) -> c_1(g^#(m, 0()))
    , 2: f^#(0(), n) -> c_2(g^#(0(), n))
    , 3: f^#(s(m), s(n)) ->
         c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
             f^#(m, p(m, n)),
             p^#(m, n),
             f^#(s(m), n))
    , 4: g^#(n, m) -> c_4()
    , 5: h^#(k, l, m, n) -> c_5()
    , 6: p^#(m, n) -> c_6() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(m, 0()) -> c_1(g^#(m, 0()))
  , f^#(0(), n) -> c_2(g^#(0(), n))
  , f^#(s(m), s(n)) ->
    c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
        f^#(m, p(m, n)),
        p^#(m, n),
        f^#(s(m), n)) }
Weak DPs:
  { g^#(n, m) -> c_4()
  , h^#(k, l, m, n) -> c_5()
  , p^#(m, n) -> c_6() }
Weak Trs:
  { f(m, 0()) -> g(m, 0())
  , f(0(), n) -> g(0(), n)
  , f(s(m), s(n)) -> h(m, n, f(m, p(m, n)), f(s(m), n))
  , g(n, m) -> bot()
  , h(k, l, m, n) -> bot()
  , p(m, n) -> bot() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

We estimate the number of application of {1,2} by applications of
Pre({1,2}) = {3}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(m, 0()) -> c_1(g^#(m, 0()))
    , 2: f^#(0(), n) -> c_2(g^#(0(), n))
    , 3: f^#(s(m), s(n)) ->
         c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
             f^#(m, p(m, n)),
             p^#(m, n),
             f^#(s(m), n))
    , 4: g^#(n, m) -> c_4()
    , 5: h^#(k, l, m, n) -> c_5()
    , 6: p^#(m, n) -> c_6() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(s(m), s(n)) ->
    c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
        f^#(m, p(m, n)),
        p^#(m, n),
        f^#(s(m), n)) }
Weak DPs:
  { f^#(m, 0()) -> c_1(g^#(m, 0()))
  , f^#(0(), n) -> c_2(g^#(0(), n))
  , g^#(n, m) -> c_4()
  , h^#(k, l, m, n) -> c_5()
  , p^#(m, n) -> c_6() }
Weak Trs:
  { f(m, 0()) -> g(m, 0())
  , f(0(), n) -> g(0(), n)
  , f(s(m), s(n)) -> h(m, n, f(m, p(m, n)), f(s(m), n))
  , g(n, m) -> bot()
  , h(k, l, m, n) -> bot()
  , p(m, n) -> bot() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ f^#(m, 0()) -> c_1(g^#(m, 0()))
, f^#(0(), n) -> c_2(g^#(0(), n))
, g^#(n, m) -> c_4()
, h^#(k, l, m, n) -> c_5()
, p^#(m, n) -> c_6() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(s(m), s(n)) ->
    c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
        f^#(m, p(m, n)),
        p^#(m, n),
        f^#(s(m), n)) }
Weak Trs:
  { f(m, 0()) -> g(m, 0())
  , f(0(), n) -> g(0(), n)
  , f(s(m), s(n)) -> h(m, n, f(m, p(m, n)), f(s(m), n))
  , g(n, m) -> bot()
  , h(k, l, m, n) -> bot()
  , p(m, n) -> bot() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { f^#(s(m), s(n)) ->
    c_3(h^#(m, n, f(m, p(m, n)), f(s(m), n)),
        f^#(m, p(m, n)),
        p^#(m, n),
        f^#(s(m), n)) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs: { f^#(s(m), s(n)) -> c_1(f^#(s(m), n)) }
Weak Trs:
  { f(m, 0()) -> g(m, 0())
  , f(0(), n) -> g(0(), n)
  , f(s(m), s(n)) -> h(m, n, f(m, p(m, n)), f(s(m), n))
  , g(n, m) -> bot()
  , h(k, l, m, n) -> bot()
  , p(m, n) -> bot() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs: { f^#(s(m), s(n)) -> c_1(f^#(s(m), n)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following argument positions are usable:
  Uargs(c_1) = {1}

TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).

        [s](x1) = [1] x1 + [2]
                              
  [f^#](x1, x2) = [2] x2 + [0]
                              
      [c_1](x1) = [1] x1 + [3]

This order satisfies following ordering constraints:

  [f^#(s(m), s(n))] = [2] n + [4]        
                    > [2] n + [3]        
                    = [c_1(f^#(s(m), n))]
                                         

Hurray, we answered YES(?,O(n^1))