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))
  , p(s(x)) -> x
  , f(x, y, 0()) -> max(x, y)
  , f(x, 0(), z) -> max(x, z)
  , f(0(), y, z) -> max(y, z)
  , f(s(x), s(y), s(z)) ->
    f(max(s(x), max(s(y), s(z))),
      p(min(s(x), max(s(y), s(z)))),
      min(s(x), min(s(y), s(z)))) }
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))
  , p^#(s(x)) -> c_7()
  , f^#(x, y, 0()) -> c_8(max^#(x, y))
  , f^#(x, 0(), z) -> c_9(max^#(x, z))
  , f^#(0(), y, z) -> c_10(max^#(y, z))
  , f^#(s(x), s(y), s(z)) ->
    c_11(f^#(max(s(x), max(s(y), s(z))),
             p(min(s(x), max(s(y), s(z)))),
             min(s(x), min(s(y), s(z)))),
         max^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         p^#(min(s(x), max(s(y), s(z)))),
         min^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         min^#(s(x), min(s(y), s(z))),
         min^#(s(y), s(z))) }

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))
  , p^#(s(x)) -> c_7()
  , f^#(x, y, 0()) -> c_8(max^#(x, y))
  , f^#(x, 0(), z) -> c_9(max^#(x, z))
  , f^#(0(), y, z) -> c_10(max^#(y, z))
  , f^#(s(x), s(y), s(z)) ->
    c_11(f^#(max(s(x), max(s(y), s(z))),
             p(min(s(x), max(s(y), s(z)))),
             min(s(x), min(s(y), s(z)))),
         max^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         p^#(min(s(x), max(s(y), s(z)))),
         min^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         min^#(s(x), min(s(y), s(z))),
         min^#(s(y), s(z))) }
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))
  , p(s(x)) -> x
  , f(x, y, 0()) -> max(x, y)
  , f(x, 0(), z) -> max(x, z)
  , f(0(), y, z) -> max(y, z)
  , f(s(x), s(y), s(z)) ->
    f(max(s(x), max(s(y), s(z))),
      p(min(s(x), max(s(y), s(z)))),
      min(s(x), min(s(y), s(z)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,5,7} by
applications of Pre({1,2,4,5,7}) = {3,6,8,9,10,11}. 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: p^#(s(x)) -> c_7()
    , 8: f^#(x, y, 0()) -> c_8(max^#(x, y))
    , 9: f^#(x, 0(), z) -> c_9(max^#(x, z))
    , 10: f^#(0(), y, z) -> c_10(max^#(y, z))
    , 11: f^#(s(x), s(y), s(z)) ->
          c_11(f^#(max(s(x), max(s(y), s(z))),
                   p(min(s(x), max(s(y), s(z)))),
                   min(s(x), min(s(y), s(z)))),
               max^#(s(x), max(s(y), s(z))),
               max^#(s(y), s(z)),
               p^#(min(s(x), max(s(y), s(z)))),
               min^#(s(x), max(s(y), s(z))),
               max^#(s(y), s(z)),
               min^#(s(x), min(s(y), s(z))),
               min^#(s(y), s(z))) }

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))
  , f^#(x, y, 0()) -> c_8(max^#(x, y))
  , f^#(x, 0(), z) -> c_9(max^#(x, z))
  , f^#(0(), y, z) -> c_10(max^#(y, z))
  , f^#(s(x), s(y), s(z)) ->
    c_11(f^#(max(s(x), max(s(y), s(z))),
             p(min(s(x), max(s(y), s(z)))),
             min(s(x), min(s(y), s(z)))),
         max^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         p^#(min(s(x), max(s(y), s(z)))),
         min^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         min^#(s(x), min(s(y), s(z))),
         min^#(s(y), s(z))) }
Weak DPs:
  { min^#(x, 0()) -> c_1()
  , min^#(0(), y) -> c_2()
  , max^#(x, 0()) -> c_4()
  , max^#(0(), y) -> c_5()
  , p^#(s(x)) -> c_7() }
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))
  , p(s(x)) -> x
  , f(x, y, 0()) -> max(x, y)
  , f(x, 0(), z) -> max(x, z)
  , f(0(), y, z) -> max(y, z)
  , f(s(x), s(y), s(z)) ->
    f(max(s(x), max(s(y), s(z))),
      p(min(s(x), max(s(y), s(z)))),
      min(s(x), min(s(y), s(z)))) }
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()
, p^#(s(x)) -> c_7() }

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))
  , f^#(x, y, 0()) -> c_8(max^#(x, y))
  , f^#(x, 0(), z) -> c_9(max^#(x, z))
  , f^#(0(), y, z) -> c_10(max^#(y, z))
  , f^#(s(x), s(y), s(z)) ->
    c_11(f^#(max(s(x), max(s(y), s(z))),
             p(min(s(x), max(s(y), s(z)))),
             min(s(x), min(s(y), s(z)))),
         max^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         p^#(min(s(x), max(s(y), s(z)))),
         min^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         min^#(s(x), min(s(y), s(z))),
         min^#(s(y), s(z))) }
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))
  , p(s(x)) -> x
  , f(x, y, 0()) -> max(x, y)
  , f(x, 0(), z) -> max(x, z)
  , f(0(), y, z) -> max(y, z)
  , f(s(x), s(y), s(z)) ->
    f(max(s(x), max(s(y), s(z))),
      p(min(s(x), max(s(y), s(z)))),
      min(s(x), min(s(y), s(z)))) }
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), s(z)) ->
    c_11(f^#(max(s(x), max(s(y), s(z))),
             p(min(s(x), max(s(y), s(z)))),
             min(s(x), min(s(y), s(z)))),
         max^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         p^#(min(s(x), max(s(y), s(z)))),
         min^#(s(x), max(s(y), s(z))),
         max^#(s(y), s(z)),
         min^#(s(x), min(s(y), s(z))),
         min^#(s(y), s(z))) }

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))
  , f^#(x, y, 0()) -> c_3(max^#(x, y))
  , f^#(x, 0(), z) -> c_4(max^#(x, z))
  , f^#(0(), y, z) -> c_5(max^#(y, z))
  , f^#(s(x), s(y), s(z)) ->
    c_6(f^#(max(s(x), max(s(y), s(z))),
            p(min(s(x), max(s(y), s(z)))),
            min(s(x), min(s(y), s(z)))),
        max^#(s(x), max(s(y), s(z))),
        max^#(s(y), s(z)),
        min^#(s(x), max(s(y), s(z))),
        max^#(s(y), s(z)),
        min^#(s(x), min(s(y), s(z))),
        min^#(s(y), s(z))) }
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))
  , p(s(x)) -> x
  , f(x, y, 0()) -> max(x, y)
  , f(x, 0(), z) -> max(x, z)
  , f(0(), y, z) -> max(y, z)
  , f(s(x), s(y), s(z)) ->
    f(max(s(x), max(s(y), s(z))),
      p(min(s(x), max(s(y), s(z)))),
      min(s(x), min(s(y), s(z)))) }
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))
    , 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))
  , f^#(x, y, 0()) -> c_3(max^#(x, y))
  , f^#(x, 0(), z) -> c_4(max^#(x, z))
  , f^#(0(), y, z) -> c_5(max^#(y, z))
  , f^#(s(x), s(y), s(z)) ->
    c_6(f^#(max(s(x), max(s(y), s(z))),
            p(min(s(x), max(s(y), s(z)))),
            min(s(x), min(s(y), s(z)))),
        max^#(s(x), max(s(y), s(z))),
        max^#(s(y), s(z)),
        min^#(s(x), max(s(y), s(z))),
        max^#(s(y), s(z)),
        min^#(s(x), min(s(y), s(z))),
        min^#(s(y), s(z))) }
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))
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..