MAYBE

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

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

We add following dependency tuples:

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

We estimate the number of application of {4,5,7,8,9,11,12} by
applications of Pre({4,5,7,8,9,11,12}) = {1,2,3,6,10}. Here rules
are labeled as follows:

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

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

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

{ gr^#(0(), x) -> c_4()
, gr^#(s(x), 0()) -> c_5()
, neq^#(0(), 0()) -> c_7()
, neq^#(0(), s(x)) -> c_8()
, neq^#(s(x), 0()) -> c_9()
, p^#(0()) -> c_11()
, p^#(s(x)) -> c_12() }

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

Strict DPs:
  { cond1^#(true(), x, y) -> c_1(cond2^#(gr(x, y), x, y), gr^#(x, y))
  , cond2^#(true(), x, y) ->
    c_2(cond1^#(neq(x, 0()), y, y), neq^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond1^#(neq(x, 0()), p(x), y), neq^#(x, 0()), p^#(x))
  , gr^#(s(x), s(y)) -> c_6(gr^#(x, y))
  , neq^#(s(x), s(y)) -> c_10(neq^#(x, y)) }
Weak Trs:
  { cond1(true(), x, y) -> cond2(gr(x, y), x, y)
  , cond2(true(), x, y) -> cond1(neq(x, 0()), y, y)
  , cond2(false(), x, y) -> cond1(neq(x, 0()), p(x), y)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , neq(0(), 0()) -> false()
  , neq(0(), s(x)) -> true()
  , neq(s(x), 0()) -> true()
  , neq(s(x), s(y)) -> neq(x, y)
  , p(0()) -> 0()
  , p(s(x)) -> x }
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^#(true(), x, y) ->
    c_2(cond1^#(neq(x, 0()), y, y), neq^#(x, 0()))
  , cond2^#(false(), x, y) ->
    c_3(cond1^#(neq(x, 0()), p(x), y), neq^#(x, 0()), p^#(x)) }

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

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

We replace rewrite rules by usable rules:

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

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

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

The input cannot be shown compatible

Arrrr..