MAYBE

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

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

We add following dependency tuples:

Strict DPs:
  { min^#(x, 0()) -> c_1()
  , min^#(0(), y) -> c_2()
  , min^#(s(x), s(y)) -> c_3(min^#(x, y))
  , max^#(x, 0()) -> c_4()
  , max^#(0(), y) -> c_5()
  , max^#(s(x), s(y)) -> c_6(max^#(x, y))
  , twice^#(0()) -> c_7()
  , twice^#(s(x)) -> c_8(twice^#(x))
  , -^#(x, 0()) -> c_9()
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , p^#(s(x)) -> c_11()
  , f^#(s(x), s(y)) ->
    c_12(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
         -^#(max(s(x), s(y)), min(s(x), s(y))),
         max^#(s(x), s(y)),
         min^#(s(x), s(y)),
         p^#(twice(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:
  { min^#(x, 0()) -> c_1()
  , min^#(0(), y) -> c_2()
  , min^#(s(x), s(y)) -> c_3(min^#(x, y))
  , max^#(x, 0()) -> c_4()
  , max^#(0(), y) -> c_5()
  , max^#(s(x), s(y)) -> c_6(max^#(x, y))
  , twice^#(0()) -> c_7()
  , twice^#(s(x)) -> c_8(twice^#(x))
  , -^#(x, 0()) -> c_9()
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , p^#(s(x)) -> c_11()
  , f^#(s(x), s(y)) ->
    c_12(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
         -^#(max(s(x), s(y)), min(s(x), s(y))),
         max^#(s(x), s(y)),
         min^#(s(x), s(y)),
         p^#(twice(min(x, y))),
         twice^#(min(x, y)),
         min^#(x, y)) }
Weak Trs:
  { min(x, 0()) -> 0()
  , min(0(), y) -> 0()
  , min(s(x), s(y)) -> s(min(x, y))
  , max(x, 0()) -> x
  , max(0(), y) -> y
  , max(s(x), s(y)) -> s(max(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,5,7,9,11} by
applications of Pre({1,2,4,5,7,9,11}) = {3,6,8,10,12}. Here rules
are labeled as follows:

  DPs:
    { 1: min^#(x, 0()) -> c_1()
    , 2: min^#(0(), y) -> c_2()
    , 3: min^#(s(x), s(y)) -> c_3(min^#(x, y))
    , 4: max^#(x, 0()) -> c_4()
    , 5: max^#(0(), y) -> c_5()
    , 6: max^#(s(x), s(y)) -> c_6(max^#(x, y))
    , 7: twice^#(0()) -> c_7()
    , 8: twice^#(s(x)) -> c_8(twice^#(x))
    , 9: -^#(x, 0()) -> c_9()
    , 10: -^#(s(x), s(y)) -> c_10(-^#(x, y))
    , 11: p^#(s(x)) -> c_11()
    , 12: f^#(s(x), s(y)) ->
          c_12(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
               -^#(max(s(x), s(y)), min(s(x), s(y))),
               max^#(s(x), s(y)),
               min^#(s(x), s(y)),
               p^#(twice(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:
  { min^#(s(x), s(y)) -> c_3(min^#(x, y))
  , max^#(s(x), s(y)) -> c_6(max^#(x, y))
  , twice^#(s(x)) -> c_8(twice^#(x))
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , f^#(s(x), s(y)) ->
    c_12(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
         -^#(max(s(x), s(y)), min(s(x), s(y))),
         max^#(s(x), s(y)),
         min^#(s(x), s(y)),
         p^#(twice(min(x, y))),
         twice^#(min(x, y)),
         min^#(x, y)) }
Weak DPs:
  { min^#(x, 0()) -> c_1()
  , min^#(0(), y) -> c_2()
  , max^#(x, 0()) -> c_4()
  , max^#(0(), y) -> c_5()
  , twice^#(0()) -> c_7()
  , -^#(x, 0()) -> c_9()
  , p^#(s(x)) -> c_11() }
Weak Trs:
  { min(x, 0()) -> 0()
  , min(0(), y) -> 0()
  , min(s(x), s(y)) -> s(min(x, y))
  , max(x, 0()) -> x
  , max(0(), y) -> y
  , max(s(x), s(y)) -> s(max(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(max(s(x), s(y)), min(s(x), s(y))), p(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.

{ min^#(x, 0()) -> c_1()
, min^#(0(), y) -> c_2()
, max^#(x, 0()) -> c_4()
, max^#(0(), y) -> c_5()
, twice^#(0()) -> c_7()
, -^#(x, 0()) -> c_9()
, p^#(s(x)) -> c_11() }

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

Strict DPs:
  { min^#(s(x), s(y)) -> c_3(min^#(x, y))
  , max^#(s(x), s(y)) -> c_6(max^#(x, y))
  , twice^#(s(x)) -> c_8(twice^#(x))
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , f^#(s(x), s(y)) ->
    c_12(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
         -^#(max(s(x), s(y)), min(s(x), s(y))),
         max^#(s(x), s(y)),
         min^#(s(x), s(y)),
         p^#(twice(min(x, y))),
         twice^#(min(x, y)),
         min^#(x, y)) }
Weak Trs:
  { min(x, 0()) -> 0()
  , min(0(), y) -> 0()
  , min(s(x), s(y)) -> s(min(x, y))
  , max(x, 0()) -> x
  , max(0(), y) -> y
  , max(s(x), s(y)) -> s(max(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { f^#(s(x), s(y)) ->
    c_12(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
         -^#(max(s(x), s(y)), min(s(x), s(y))),
         max^#(s(x), s(y)),
         min^#(s(x), s(y)),
         p^#(twice(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:
  { min^#(s(x), s(y)) -> c_1(min^#(x, y))
  , max^#(s(x), s(y)) -> c_2(max^#(x, y))
  , twice^#(s(x)) -> c_3(twice^#(x))
  , -^#(s(x), s(y)) -> c_4(-^#(x, y))
  , f^#(s(x), s(y)) ->
    c_5(f^#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))),
        -^#(max(s(x), s(y)), min(s(x), s(y))),
        max^#(s(x), s(y)),
        min^#(s(x), s(y)),
        twice^#(min(x, y)),
        min^#(x, y)) }
Weak Trs:
  { min(x, 0()) -> 0()
  , min(0(), y) -> 0()
  , min(s(x), s(y)) -> s(min(x, y))
  , max(x, 0()) -> x
  , max(0(), y) -> y
  , max(s(x), s(y)) -> s(max(x, y))
  , twice(0()) -> 0()
  , twice(s(x)) -> s(s(twice(x)))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

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

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

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

The input cannot be shown compatible

Arrrr..