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

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

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))
  , minus^#(s(x), s(y)) -> c_8(minus^#(x, any(y)), any^#(y))
  , any^#(s(x)) -> c_10(any^#(x))
  , gcd^#(s(x), s(y)) ->
    c_11(gcd^#(minus(max(x, y), min(x, y)), s(min(x, y))),
         minus^#(max(x, y), min(x, y)),
         max^#(x, y),
         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))
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> s(minus(x, any(y)))
  , any(x) -> x
  , any(s(x)) -> s(s(any(x)))
  , gcd(s(x), s(y)) ->
    gcd(minus(max(x, y), min(x, y)), s(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))
    , minus(x, 0()) -> x
    , minus(s(x), s(y)) -> s(minus(x, any(y)))
    , any(x) -> x
    , any(s(x)) -> s(s(any(x))) }

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))
  , minus^#(s(x), s(y)) -> c_8(minus^#(x, any(y)), any^#(y))
  , any^#(s(x)) -> c_10(any^#(x))
  , gcd^#(s(x), s(y)) ->
    c_11(gcd^#(minus(max(x, y), min(x, y)), s(min(x, y))),
         minus^#(max(x, y), min(x, y)),
         max^#(x, y),
         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))
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> s(minus(x, any(y)))
  , any(x) -> x
  , any(s(x)) -> s(s(any(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..