MAYBE

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

Strict Trs:
  { cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
  , cond2(true(), x, y) -> cond1(y, y)
  , cond2(false(), x, y) -> cond1(p(x), y)
  , gr(s(x), s(y)) -> gr(x, y)
  , gr(s(x), 0()) -> true()
  , gr(0(), x) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , neq(s(x), s(y)) -> neq(x, y)
  , neq(s(x), 0()) -> true()
  , neq(0(), s(x)) -> true()
  , neq(0(), 0()) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { cond1^#(s(x), y) ->
    c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
  , cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
  , cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y), p^#(x))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(0(), x) -> c_6()
  , p^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
  , neq^#(s(x), 0()) -> c_10()
  , neq^#(0(), s(x)) -> c_11()
  , neq^#(0(), 0()) -> c_12() }

and mark the set of starting terms.

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

Strict DPs:
  { cond1^#(s(x), y) ->
    c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
  , cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
  , cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y), p^#(x))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , gr^#(s(x), 0()) -> c_5()
  , gr^#(0(), x) -> c_6()
  , p^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
  , neq^#(s(x), 0()) -> c_10()
  , neq^#(0(), s(x)) -> c_11()
  , neq^#(0(), 0()) -> c_12() }
Weak Trs:
  { cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
  , cond2(true(), x, y) -> cond1(y, y)
  , cond2(false(), x, y) -> cond1(p(x), y)
  , gr(s(x), s(y)) -> gr(x, y)
  , gr(s(x), 0()) -> true()
  , gr(0(), x) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , neq(s(x), s(y)) -> neq(x, y)
  , neq(s(x), 0()) -> true()
  , neq(0(), s(x)) -> true()
  , neq(0(), 0()) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {5,6,7,8,10,11,12} by
applications of Pre({5,6,7,8,10,11,12}) = {1,3,4,9}. Here rules are
labeled as follows:

  DPs:
    { 1: cond1^#(s(x), y) ->
         c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
    , 2: cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
    , 3: cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y), p^#(x))
    , 4: gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
    , 5: gr^#(s(x), 0()) -> c_5()
    , 6: gr^#(0(), x) -> c_6()
    , 7: p^#(s(x)) -> c_7()
    , 8: p^#(0()) -> c_8()
    , 9: neq^#(s(x), s(y)) -> c_9(neq^#(x, y))
    , 10: neq^#(s(x), 0()) -> c_10()
    , 11: neq^#(0(), s(x)) -> c_11()
    , 12: neq^#(0(), 0()) -> c_12() }

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

Strict DPs:
  { cond1^#(s(x), y) ->
    c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
  , cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
  , cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y), p^#(x))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , neq^#(s(x), s(y)) -> c_9(neq^#(x, y)) }
Weak DPs:
  { gr^#(s(x), 0()) -> c_5()
  , gr^#(0(), x) -> c_6()
  , p^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , neq^#(s(x), 0()) -> c_10()
  , neq^#(0(), s(x)) -> c_11()
  , neq^#(0(), 0()) -> c_12() }
Weak Trs:
  { cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
  , cond2(true(), x, y) -> cond1(y, y)
  , cond2(false(), x, y) -> cond1(p(x), y)
  , gr(s(x), s(y)) -> gr(x, y)
  , gr(s(x), 0()) -> true()
  , gr(0(), x) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , neq(s(x), s(y)) -> neq(x, y)
  , neq(s(x), 0()) -> true()
  , neq(0(), s(x)) -> true()
  , neq(0(), 0()) -> false() }
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.

{ gr^#(s(x), 0()) -> c_5()
, gr^#(0(), x) -> c_6()
, p^#(s(x)) -> c_7()
, p^#(0()) -> c_8()
, neq^#(s(x), 0()) -> c_10()
, neq^#(0(), s(x)) -> c_11()
, neq^#(0(), 0()) -> c_12() }

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

Strict DPs:
  { cond1^#(s(x), y) ->
    c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
  , cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
  , cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y), p^#(x))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , neq^#(s(x), s(y)) -> c_9(neq^#(x, y)) }
Weak Trs:
  { cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
  , cond2(true(), x, y) -> cond1(y, y)
  , cond2(false(), x, y) -> cond1(p(x), y)
  , gr(s(x), s(y)) -> gr(x, y)
  , gr(s(x), 0()) -> true()
  , gr(0(), x) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , neq(s(x), s(y)) -> neq(x, y)
  , neq(s(x), 0()) -> true()
  , neq(0(), s(x)) -> true()
  , neq(0(), 0()) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y), p^#(x)) }

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

Strict DPs:
  { cond1^#(s(x), y) ->
    c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
  , cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
  , cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , neq^#(s(x), s(y)) -> c_5(neq^#(x, y)) }
Weak Trs:
  { cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
  , cond2(true(), x, y) -> cond1(y, y)
  , cond2(false(), x, y) -> cond1(p(x), y)
  , gr(s(x), s(y)) -> gr(x, y)
  , gr(s(x), 0()) -> true()
  , gr(0(), x) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , neq(s(x), s(y)) -> neq(x, y)
  , neq(s(x), 0()) -> true()
  , neq(0(), s(x)) -> true()
  , neq(0(), 0()) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { cond1^#(s(x), y) ->
    c_1(cond2^#(gr(s(x), y), s(x), y), gr^#(s(x), y))
  , cond2^#(true(), x, y) -> c_2(cond1^#(y, y))
  , cond2^#(false(), x, y) -> c_3(cond1^#(p(x), y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , neq^#(s(x), s(y)) -> c_5(neq^#(x, y)) }
Weak Trs:
  { gr(s(x), s(y)) -> gr(x, y)
  , gr(s(x), 0()) -> true()
  , gr(0(), x) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..