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))
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(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))
  , +^#(0(), y) -> c_7()
  , +^#(s(x), y) -> c_8(+^#(x, y))
  , -^#(x, 0()) -> c_9()
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , *^#(x, 0()) -> c_11()
  , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y))
  , p^#(s(x)) -> c_13()
  , f^#(s(x), s(y)) ->
    c_14(f^#(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))),
         -^#(min(s(x), s(y)), max(s(x), s(y))),
         min^#(s(x), s(y)),
         max^#(s(x), s(y)),
         *^#(s(x), s(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))
  , +^#(0(), y) -> c_7()
  , +^#(s(x), y) -> c_8(+^#(x, y))
  , -^#(x, 0()) -> c_9()
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , *^#(x, 0()) -> c_11()
  , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y))
  , p^#(s(x)) -> c_13()
  , f^#(s(x), s(y)) ->
    c_14(f^#(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))),
         -^#(min(s(x), s(y)), max(s(x), s(y))),
         min^#(s(x), s(y)),
         max^#(s(x), s(y)),
         *^#(s(x), s(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))
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,5,7,9,11,13} by
applications of Pre({1,2,4,5,7,9,11,13}) = {3,6,8,10,12,14}. 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: +^#(0(), y) -> c_7()
    , 8: +^#(s(x), y) -> c_8(+^#(x, y))
    , 9: -^#(x, 0()) -> c_9()
    , 10: -^#(s(x), s(y)) -> c_10(-^#(x, y))
    , 11: *^#(x, 0()) -> c_11()
    , 12: *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y))
    , 13: p^#(s(x)) -> c_13()
    , 14: f^#(s(x), s(y)) ->
          c_14(f^#(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))),
               -^#(min(s(x), s(y)), max(s(x), s(y))),
               min^#(s(x), s(y)),
               max^#(s(x), s(y)),
               *^#(s(x), s(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))
  , +^#(s(x), y) -> c_8(+^#(x, y))
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y))
  , f^#(s(x), s(y)) ->
    c_14(f^#(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))),
         -^#(min(s(x), s(y)), max(s(x), s(y))),
         min^#(s(x), s(y)),
         max^#(s(x), s(y)),
         *^#(s(x), s(y))) }
Weak DPs:
  { min^#(x, 0()) -> c_1()
  , min^#(0(), y) -> c_2()
  , max^#(x, 0()) -> c_4()
  , max^#(0(), y) -> c_5()
  , +^#(0(), y) -> c_7()
  , -^#(x, 0()) -> c_9()
  , *^#(x, 0()) -> c_11()
  , p^#(s(x)) -> c_13() }
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))
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(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()
, +^#(0(), y) -> c_7()
, -^#(x, 0()) -> c_9()
, *^#(x, 0()) -> c_11()
, p^#(s(x)) -> c_13() }

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))
  , +^#(s(x), y) -> c_8(+^#(x, y))
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y))
  , f^#(s(x), s(y)) ->
    c_14(f^#(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))),
         -^#(min(s(x), s(y)), max(s(x), s(y))),
         min^#(s(x), s(y)),
         max^#(s(x), s(y)),
         *^#(s(x), s(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))
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , p(s(x)) -> x
  , f(s(x), s(y)) ->
    f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(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))
    , +(0(), y) -> y
    , +(s(x), y) -> s(+(x, y))
    , -(x, 0()) -> x
    , -(s(x), s(y)) -> -(x, y)
    , *(x, 0()) -> 0()
    , *(x, s(y)) -> +(x, *(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))
  , +^#(s(x), y) -> c_8(+^#(x, y))
  , -^#(s(x), s(y)) -> c_10(-^#(x, y))
  , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y))
  , f^#(s(x), s(y)) ->
    c_14(f^#(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))),
         -^#(min(s(x), s(y)), max(s(x), s(y))),
         min^#(s(x), s(y)),
         max^#(s(x), s(y)),
         *^#(s(x), s(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))
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..