MAYBE

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

Strict Trs:
  { +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , f(s(x)) ->
    f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { +^#(0(), y) -> c_1()
  , +^#(s(x), y) -> c_2(+^#(x, y))
  , *^#(x, 0()) -> c_3()
  , *^#(x, s(y)) -> c_4(+^#(x, *(x, y)), *^#(x, y))
  , twice^#(0()) -> c_5()
  , twice^#(s(x)) -> c_6(twice^#(x))
  , -^#(x, 0()) -> c_7()
  , -^#(s(x), s(y)) -> c_8(-^#(x, y))
  , f^#(s(x)) ->
    c_9(f^#(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))),
        -^#(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0())))),
        *^#(s(s(x)), s(s(x))),
        +^#(*(s(x), s(s(x))), s(s(0()))),
        *^#(s(x), s(s(x)))) }

and mark the set of starting terms.

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

Strict DPs:
  { +^#(0(), y) -> c_1()
  , +^#(s(x), y) -> c_2(+^#(x, y))
  , *^#(x, 0()) -> c_3()
  , *^#(x, s(y)) -> c_4(+^#(x, *(x, y)), *^#(x, y))
  , twice^#(0()) -> c_5()
  , twice^#(s(x)) -> c_6(twice^#(x))
  , -^#(x, 0()) -> c_7()
  , -^#(s(x), s(y)) -> c_8(-^#(x, y))
  , f^#(s(x)) ->
    c_9(f^#(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))),
        -^#(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0())))),
        *^#(s(s(x)), s(s(x))),
        +^#(*(s(x), s(s(x))), s(s(0()))),
        *^#(s(x), s(s(x)))) }
Weak Trs:
  { +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , f(s(x)) ->
    f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,7} by applications
of Pre({1,3,5,7}) = {2,4,6,8,9}. Here rules are labeled as follows:

  DPs:
    { 1: +^#(0(), y) -> c_1()
    , 2: +^#(s(x), y) -> c_2(+^#(x, y))
    , 3: *^#(x, 0()) -> c_3()
    , 4: *^#(x, s(y)) -> c_4(+^#(x, *(x, y)), *^#(x, y))
    , 5: twice^#(0()) -> c_5()
    , 6: twice^#(s(x)) -> c_6(twice^#(x))
    , 7: -^#(x, 0()) -> c_7()
    , 8: -^#(s(x), s(y)) -> c_8(-^#(x, y))
    , 9: f^#(s(x)) ->
         c_9(f^#(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))),
             -^#(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0())))),
             *^#(s(s(x)), s(s(x))),
             +^#(*(s(x), s(s(x))), s(s(0()))),
             *^#(s(x), s(s(x)))) }

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

Strict DPs:
  { +^#(s(x), y) -> c_2(+^#(x, y))
  , *^#(x, s(y)) -> c_4(+^#(x, *(x, y)), *^#(x, y))
  , twice^#(s(x)) -> c_6(twice^#(x))
  , -^#(s(x), s(y)) -> c_8(-^#(x, y))
  , f^#(s(x)) ->
    c_9(f^#(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))),
        -^#(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0())))),
        *^#(s(s(x)), s(s(x))),
        +^#(*(s(x), s(s(x))), s(s(0()))),
        *^#(s(x), s(s(x)))) }
Weak DPs:
  { +^#(0(), y) -> c_1()
  , *^#(x, 0()) -> c_3()
  , twice^#(0()) -> c_5()
  , -^#(x, 0()) -> c_7() }
Weak Trs:
  { +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , f(s(x)) ->
    f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(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.

{ +^#(0(), y) -> c_1()
, *^#(x, 0()) -> c_3()
, twice^#(0()) -> c_5()
, -^#(x, 0()) -> c_7() }

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

Strict DPs:
  { +^#(s(x), y) -> c_2(+^#(x, y))
  , *^#(x, s(y)) -> c_4(+^#(x, *(x, y)), *^#(x, y))
  , twice^#(s(x)) -> c_6(twice^#(x))
  , -^#(s(x), s(y)) -> c_8(-^#(x, y))
  , f^#(s(x)) ->
    c_9(f^#(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))),
        -^#(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0())))),
        *^#(s(s(x)), s(s(x))),
        +^#(*(s(x), s(s(x))), s(s(0()))),
        *^#(s(x), s(s(x)))) }
Weak Trs:
  { +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , f(s(x)) ->
    f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { +(0(), y) -> y
    , +(s(x), y) -> s(+(x, y))
    , *(x, 0()) -> 0()
    , *(x, s(y)) -> +(x, *(x, y))
    , -(x, 0()) -> x
    , -(s(x), s(y)) -> -(x, y) }

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

Strict DPs:
  { +^#(s(x), y) -> c_2(+^#(x, y))
  , *^#(x, s(y)) -> c_4(+^#(x, *(x, y)), *^#(x, y))
  , twice^#(s(x)) -> c_6(twice^#(x))
  , -^#(s(x), s(y)) -> c_8(-^#(x, y))
  , f^#(s(x)) ->
    c_9(f^#(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0()))))),
        -^#(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0())))),
        *^#(s(s(x)), s(s(x))),
        +^#(*(s(x), s(s(x))), s(s(0()))),
        *^#(s(x), s(s(x)))) }
Weak Trs:
  { +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..