MAYBE

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

Strict Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , gcd(0(), y) -> y
  , gcd(s(x), 0()) -> s(x)
  , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
  , if_gcd(true(), x, y) -> gcd(minus(x, y), y)
  , if_gcd(false(), x, y) -> gcd(minus(y, x), x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), x) -> c_5()
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , gcd^#(0(), y) -> c_7()
  , gcd^#(s(x), 0()) -> c_8()
  , gcd^#(s(x), s(y)) ->
    c_9(if_gcd^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_gcd^#(true(), x, y) ->
    c_10(gcd^#(minus(x, y), y), minus^#(x, y))
  , if_gcd^#(false(), x, y) ->
    c_11(gcd^#(minus(y, x), x), minus^#(y, x)) }

and mark the set of starting terms.

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

Strict DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), x) -> c_5()
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , gcd^#(0(), y) -> c_7()
  , gcd^#(s(x), 0()) -> c_8()
  , gcd^#(s(x), s(y)) ->
    c_9(if_gcd^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_gcd^#(true(), x, y) ->
    c_10(gcd^#(minus(x, y), y), minus^#(x, y))
  , if_gcd^#(false(), x, y) ->
    c_11(gcd^#(minus(y, x), x), minus^#(y, x)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , gcd(0(), y) -> y
  , gcd(s(x), 0()) -> s(x)
  , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
  , if_gcd(true(), x, y) -> gcd(minus(x, y), y)
  , if_gcd(false(), x, y) -> gcd(minus(y, x), x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: le^#(0(), y) -> c_1()
    , 2: le^#(s(x), 0()) -> c_2()
    , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y))
    , 4: minus^#(x, 0()) -> c_4()
    , 5: minus^#(0(), x) -> c_5()
    , 6: minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
    , 7: gcd^#(0(), y) -> c_7()
    , 8: gcd^#(s(x), 0()) -> c_8()
    , 9: gcd^#(s(x), s(y)) ->
         c_9(if_gcd^#(le(y, x), s(x), s(y)), le^#(y, x))
    , 10: if_gcd^#(true(), x, y) ->
          c_10(gcd^#(minus(x, y), y), minus^#(x, y))
    , 11: if_gcd^#(false(), x, y) ->
          c_11(gcd^#(minus(y, x), x), minus^#(y, x)) }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , gcd^#(s(x), s(y)) ->
    c_9(if_gcd^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_gcd^#(true(), x, y) ->
    c_10(gcd^#(minus(x, y), y), minus^#(x, y))
  , if_gcd^#(false(), x, y) ->
    c_11(gcd^#(minus(y, x), x), minus^#(y, x)) }
Weak DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), x) -> c_5()
  , gcd^#(0(), y) -> c_7()
  , gcd^#(s(x), 0()) -> c_8() }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , gcd(0(), y) -> y
  , gcd(s(x), 0()) -> s(x)
  , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
  , if_gcd(true(), x, y) -> gcd(minus(x, y), y)
  , if_gcd(false(), x, y) -> gcd(minus(y, x), x) }
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.

{ le^#(0(), y) -> c_1()
, le^#(s(x), 0()) -> c_2()
, minus^#(x, 0()) -> c_4()
, minus^#(0(), x) -> c_5()
, gcd^#(0(), y) -> c_7()
, gcd^#(s(x), 0()) -> c_8() }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , gcd^#(s(x), s(y)) ->
    c_9(if_gcd^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_gcd^#(true(), x, y) ->
    c_10(gcd^#(minus(x, y), y), minus^#(x, y))
  , if_gcd^#(false(), x, y) ->
    c_11(gcd^#(minus(y, x), x), minus^#(y, x)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , gcd(0(), y) -> y
  , gcd(s(x), 0()) -> s(x)
  , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
  , if_gcd(true(), x, y) -> gcd(minus(x, y), y)
  , if_gcd(false(), x, y) -> gcd(minus(y, x), x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { le(0(), y) -> true()
    , le(s(x), 0()) -> false()
    , le(s(x), s(y)) -> le(x, y)
    , minus(x, 0()) -> x
    , minus(0(), x) -> 0()
    , minus(s(x), s(y)) -> minus(x, y) }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , gcd^#(s(x), s(y)) ->
    c_9(if_gcd^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_gcd^#(true(), x, y) ->
    c_10(gcd^#(minus(x, y), y), minus^#(x, y))
  , if_gcd^#(false(), x, y) ->
    c_11(gcd^#(minus(y, x), x), minus^#(y, x)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..