MAYBE

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

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

We add following dependency tuples:

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

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

  DPs:
    { 1: cond^#(true(), x) ->
         c_1(cond^#(and(even(x), gr(x, 0())), p(x)),
             and^#(even(x), gr(x, 0())),
             even^#(x),
             gr^#(x, 0()),
             p^#(x))
    , 2: and^#(x, false()) -> c_2()
    , 3: and^#(true(), true()) -> c_3()
    , 4: and^#(false(), x) -> c_4()
    , 5: even^#(0()) -> c_5()
    , 6: even^#(s(0())) -> c_6()
    , 7: even^#(s(s(x))) -> c_7(even^#(x))
    , 8: gr^#(0(), x) -> c_8()
    , 9: gr^#(s(x), 0()) -> c_9()
    , 10: gr^#(s(x), s(y())) -> c_10(gr^#(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:
  { cond^#(true(), x) ->
    c_1(cond^#(and(even(x), gr(x, 0())), p(x)),
        and^#(even(x), gr(x, 0())),
        even^#(x),
        gr^#(x, 0()),
        p^#(x))
  , even^#(s(s(x))) -> c_7(even^#(x))
  , gr^#(s(x), s(y())) -> c_10(gr^#(x, y())) }
Weak DPs:
  { and^#(x, false()) -> c_2()
  , and^#(true(), true()) -> c_3()
  , and^#(false(), x) -> c_4()
  , even^#(0()) -> c_5()
  , even^#(s(0())) -> c_6()
  , gr^#(0(), x) -> c_8()
  , gr^#(s(x), 0()) -> c_9()
  , p^#(0()) -> c_11()
  , p^#(s(x)) -> c_12() }
Weak Trs:
  { cond(true(), x) -> cond(and(even(x), gr(x, 0())), p(x))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y())) -> gr(x, y())
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {3} by applications of
Pre({3}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: cond^#(true(), x) ->
         c_1(cond^#(and(even(x), gr(x, 0())), p(x)),
             and^#(even(x), gr(x, 0())),
             even^#(x),
             gr^#(x, 0()),
             p^#(x))
    , 2: even^#(s(s(x))) -> c_7(even^#(x))
    , 3: gr^#(s(x), s(y())) -> c_10(gr^#(x, y()))
    , 4: and^#(x, false()) -> c_2()
    , 5: and^#(true(), true()) -> c_3()
    , 6: and^#(false(), x) -> c_4()
    , 7: even^#(0()) -> c_5()
    , 8: even^#(s(0())) -> c_6()
    , 9: gr^#(0(), x) -> c_8()
    , 10: gr^#(s(x), 0()) -> c_9()
    , 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:
  { cond^#(true(), x) ->
    c_1(cond^#(and(even(x), gr(x, 0())), p(x)),
        and^#(even(x), gr(x, 0())),
        even^#(x),
        gr^#(x, 0()),
        p^#(x))
  , even^#(s(s(x))) -> c_7(even^#(x)) }
Weak DPs:
  { and^#(x, false()) -> c_2()
  , and^#(true(), true()) -> c_3()
  , and^#(false(), x) -> c_4()
  , even^#(0()) -> c_5()
  , even^#(s(0())) -> c_6()
  , gr^#(0(), x) -> c_8()
  , gr^#(s(x), 0()) -> c_9()
  , gr^#(s(x), s(y())) -> c_10(gr^#(x, y()))
  , p^#(0()) -> c_11()
  , p^#(s(x)) -> c_12() }
Weak Trs:
  { cond(true(), x) -> cond(and(even(x), gr(x, 0())), p(x))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y())) -> gr(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.

{ and^#(x, false()) -> c_2()
, and^#(true(), true()) -> c_3()
, and^#(false(), x) -> c_4()
, even^#(0()) -> c_5()
, even^#(s(0())) -> c_6()
, gr^#(0(), x) -> c_8()
, gr^#(s(x), 0()) -> c_9()
, gr^#(s(x), s(y())) -> c_10(gr^#(x, y()))
, p^#(0()) -> c_11()
, p^#(s(x)) -> c_12() }

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

Strict DPs:
  { cond^#(true(), x) ->
    c_1(cond^#(and(even(x), gr(x, 0())), p(x)),
        and^#(even(x), gr(x, 0())),
        even^#(x),
        gr^#(x, 0()),
        p^#(x))
  , even^#(s(s(x))) -> c_7(even^#(x)) }
Weak Trs:
  { cond(true(), x) -> cond(and(even(x), gr(x, 0())), p(x))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y())) -> gr(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:

  { cond^#(true(), x) ->
    c_1(cond^#(and(even(x), gr(x, 0())), p(x)),
        and^#(even(x), gr(x, 0())),
        even^#(x),
        gr^#(x, 0()),
        p^#(x)) }

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

Strict DPs:
  { cond^#(true(), x) ->
    c_1(cond^#(and(even(x), gr(x, 0())), p(x)), even^#(x))
  , even^#(s(s(x))) -> c_2(even^#(x)) }
Weak Trs:
  { cond(true(), x) -> cond(and(even(x), gr(x, 0())), p(x))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y())) -> gr(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:
    { and(x, false()) -> false()
    , and(true(), true()) -> true()
    , and(false(), x) -> false()
    , even(0()) -> true()
    , even(s(0())) -> false()
    , even(s(s(x))) -> even(x)
    , gr(0(), x) -> false()
    , gr(s(x), 0()) -> true()
    , gr(s(x), s(y())) -> gr(x, y())
    , p(0()) -> 0()
    , p(s(x)) -> x }

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

Strict DPs:
  { cond^#(true(), x) ->
    c_1(cond^#(and(even(x), gr(x, 0())), p(x)), even^#(x))
  , even^#(s(s(x))) -> c_2(even^#(x)) }
Weak Trs:
  { and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y())) -> gr(x, y())
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..