MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { U11(tt(), M, N) -> U12(tt(), M, N)
  , U12(tt(), M, N) -> s(plus(N, M))
  , plus(N, s(M)) -> U11(tt(), M, N)
  , plus(N, 0()) -> N
  , U21(tt(), M, N) -> U22(tt(), M, N)
  , U22(tt(), M, N) -> plus(x(N, M), N)
  , x(N, s(M)) -> U21(tt(), M, N)
  , x(N, 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { U11^#(tt(), M, N) -> c_1(U12^#(tt(), M, N))
  , U12^#(tt(), M, N) -> c_2(plus^#(N, M))
  , plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
  , plus^#(N, 0()) -> c_4()
  , U21^#(tt(), M, N) -> c_5(U22^#(tt(), M, N))
  , U22^#(tt(), M, N) -> c_6(plus^#(x(N, M), N), x^#(N, M))
  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
  , x^#(N, 0()) -> c_8() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { U11^#(tt(), M, N) -> c_1(U12^#(tt(), M, N))
  , U12^#(tt(), M, N) -> c_2(plus^#(N, M))
  , plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
  , plus^#(N, 0()) -> c_4()
  , U21^#(tt(), M, N) -> c_5(U22^#(tt(), M, N))
  , U22^#(tt(), M, N) -> c_6(plus^#(x(N, M), N), x^#(N, M))
  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
  , x^#(N, 0()) -> c_8() }
Weak Trs:
  { U11(tt(), M, N) -> U12(tt(), M, N)
  , U12(tt(), M, N) -> s(plus(N, M))
  , plus(N, s(M)) -> U11(tt(), M, N)
  , plus(N, 0()) -> N
  , U21(tt(), M, N) -> U22(tt(), M, N)
  , U22(tt(), M, N) -> plus(x(N, M), N)
  , x(N, s(M)) -> U21(tt(), M, N)
  , x(N, 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {4,8} by applications of
Pre({4,8}) = {2,6}. Here rules are labeled as follows:

  DPs:
    { 1: U11^#(tt(), M, N) -> c_1(U12^#(tt(), M, N))
    , 2: U12^#(tt(), M, N) -> c_2(plus^#(N, M))
    , 3: plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
    , 4: plus^#(N, 0()) -> c_4()
    , 5: U21^#(tt(), M, N) -> c_5(U22^#(tt(), M, N))
    , 6: U22^#(tt(), M, N) -> c_6(plus^#(x(N, M), N), x^#(N, M))
    , 7: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
    , 8: x^#(N, 0()) -> c_8() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { U11^#(tt(), M, N) -> c_1(U12^#(tt(), M, N))
  , U12^#(tt(), M, N) -> c_2(plus^#(N, M))
  , plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
  , U21^#(tt(), M, N) -> c_5(U22^#(tt(), M, N))
  , U22^#(tt(), M, N) -> c_6(plus^#(x(N, M), N), x^#(N, M))
  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N)) }
Weak DPs:
  { plus^#(N, 0()) -> c_4()
  , x^#(N, 0()) -> c_8() }
Weak Trs:
  { U11(tt(), M, N) -> U12(tt(), M, N)
  , U12(tt(), M, N) -> s(plus(N, M))
  , plus(N, s(M)) -> U11(tt(), M, N)
  , plus(N, 0()) -> N
  , U21(tt(), M, N) -> U22(tt(), M, N)
  , U22(tt(), M, N) -> plus(x(N, M), N)
  , x(N, s(M)) -> U21(tt(), M, N)
  , x(N, 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

{ plus^#(N, 0()) -> c_4()
, x^#(N, 0()) -> c_8() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { U11^#(tt(), M, N) -> c_1(U12^#(tt(), M, N))
  , U12^#(tt(), M, N) -> c_2(plus^#(N, M))
  , plus^#(N, s(M)) -> c_3(U11^#(tt(), M, N))
  , U21^#(tt(), M, N) -> c_5(U22^#(tt(), M, N))
  , U22^#(tt(), M, N) -> c_6(plus^#(x(N, M), N), x^#(N, M))
  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N)) }
Weak Trs:
  { U11(tt(), M, N) -> U12(tt(), M, N)
  , U12(tt(), M, N) -> s(plus(N, M))
  , plus(N, s(M)) -> U11(tt(), M, N)
  , plus(N, 0()) -> N
  , U21(tt(), M, N) -> U22(tt(), M, N)
  , U22(tt(), M, N) -> plus(x(N, M), N)
  , x(N, s(M)) -> U21(tt(), M, N)
  , x(N, 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..