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

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

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), s(y)) -> c_8(-^#(x, y))
  , gcd^#(x, s(y), s(z)) ->
    c_10(gcd^#(x, -(max(y, z), min(y, z)), s(min(y, z))),
         -^#(max(y, z), min(y, z)),
         max^#(y, z),
         min^#(y, z),
         min^#(y, z))
  , gcd^#(s(x), y, s(z)) ->
    c_13(gcd^#(-(max(x, z), min(x, z)), y, s(min(x, z))),
         -^#(max(x, z), min(x, z)),
         max^#(x, z),
         min^#(x, z),
         min^#(x, z))
  , gcd^#(s(x), s(y), z) ->
    c_14(gcd^#(-(max(x, y), min(x, y)), s(min(x, y)), z),
         -^#(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))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , gcd(x, 0(), 0()) -> x
  , gcd(x, s(y), s(z)) ->
    gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
  , gcd(0(), y, 0()) -> y
  , gcd(0(), 0(), z) -> z
  , gcd(s(x), y, s(z)) ->
    gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
  , gcd(s(x), s(y), z) ->
    gcd(-(max(x, y), min(x, y)), s(min(x, y)), 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))
    , -(x, 0()) -> x
    , -(s(x), s(y)) -> -(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), s(y)) -> c_8(-^#(x, y))
  , gcd^#(x, s(y), s(z)) ->
    c_10(gcd^#(x, -(max(y, z), min(y, z)), s(min(y, z))),
         -^#(max(y, z), min(y, z)),
         max^#(y, z),
         min^#(y, z),
         min^#(y, z))
  , gcd^#(s(x), y, s(z)) ->
    c_13(gcd^#(-(max(x, z), min(x, z)), y, s(min(x, z))),
         -^#(max(x, z), min(x, z)),
         max^#(x, z),
         min^#(x, z),
         min^#(x, z))
  , gcd^#(s(x), s(y), z) ->
    c_14(gcd^#(-(max(x, y), min(x, y)), s(min(x, y)), z),
         -^#(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))
  , -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..