MAYBE

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

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

We add following dependency tuples:

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

and mark the set of starting terms.

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

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

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

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

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

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

{ -^#(x, 0()) -> c_1()
, min^#(x, 0()) -> c_3()
, min^#(0(), y) -> c_4()
, twice^#(0()) -> c_6() }

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

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

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { -(x, 0()) -> x
    , -(s(x), s(y)) -> -(x, y)
    , min(x, 0()) -> 0()
    , min(0(), y) -> 0()
    , min(s(x), s(y)) -> s(min(x, y))
    , twice(0()) -> 0()
    , twice(s(x)) -> s(s(twice(x))) }

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

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

The input cannot be shown compatible

Arrrr..