MAYBE

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

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

We add following dependency tuples:

Strict DPs:
  { cond^#(true(), x, y) ->
    c_1(cond^#(gr(x, y), x, add(x, y)), gr^#(x, y), add^#(x, y))
  , gr^#(0(), x) -> c_2()
  , gr^#(s(x), 0()) -> c_3()
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , add^#(0(), x) -> c_5()
  , add^#(s(x), y) -> c_6(add^#(x, y)) }

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, y) ->
    c_1(cond^#(gr(x, y), x, add(x, y)), gr^#(x, y), add^#(x, y))
  , gr^#(0(), x) -> c_2()
  , gr^#(s(x), 0()) -> c_3()
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , add^#(0(), x) -> c_5()
  , add^#(s(x), y) -> c_6(add^#(x, y)) }
Weak Trs:
  { cond(true(), x, y) -> cond(gr(x, y), x, add(x, y))
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , add(0(), x) -> x
  , add(s(x), y) -> s(add(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: cond^#(true(), x, y) ->
         c_1(cond^#(gr(x, y), x, add(x, y)), gr^#(x, y), add^#(x, y))
    , 2: gr^#(0(), x) -> c_2()
    , 3: gr^#(s(x), 0()) -> c_3()
    , 4: gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
    , 5: add^#(0(), x) -> c_5()
    , 6: add^#(s(x), y) -> c_6(add^#(x, y)) }

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

Strict DPs:
  { cond^#(true(), x, y) ->
    c_1(cond^#(gr(x, y), x, add(x, y)), gr^#(x, y), add^#(x, y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , add^#(s(x), y) -> c_6(add^#(x, y)) }
Weak DPs:
  { gr^#(0(), x) -> c_2()
  , gr^#(s(x), 0()) -> c_3()
  , add^#(0(), x) -> c_5() }
Weak Trs:
  { cond(true(), x, y) -> cond(gr(x, y), x, add(x, y))
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , add(0(), x) -> x
  , add(s(x), y) -> s(add(x, y)) }
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_2()
, gr^#(s(x), 0()) -> c_3()
, add^#(0(), x) -> c_5() }

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

Strict DPs:
  { cond^#(true(), x, y) ->
    c_1(cond^#(gr(x, y), x, add(x, y)), gr^#(x, y), add^#(x, y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , add^#(s(x), y) -> c_6(add^#(x, y)) }
Weak Trs:
  { cond(true(), x, y) -> cond(gr(x, y), x, add(x, y))
  , gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , add(0(), x) -> x
  , add(s(x), y) -> s(add(x, y)) }
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)
    , add(0(), x) -> x
    , add(s(x), y) -> s(add(x, y)) }

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

Strict DPs:
  { cond^#(true(), x, y) ->
    c_1(cond^#(gr(x, y), x, add(x, y)), gr^#(x, y), add^#(x, y))
  , gr^#(s(x), s(y)) -> c_4(gr^#(x, y))
  , add^#(s(x), y) -> c_6(add^#(x, y)) }
Weak Trs:
  { gr(0(), x) -> false()
  , gr(s(x), 0()) -> true()
  , gr(s(x), s(y)) -> gr(x, y)
  , add(0(), x) -> x
  , add(s(x), y) -> s(add(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..