MAYBE

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

Strict Trs:
  { log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
  , cond(true(), x, y) -> s(0())
  , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
  , le(s(u), s(v)) -> le(u, v)
  , le(s(u), 0()) -> false()
  , le(0(), v) -> true()
  , double(s(x)) -> s(s(double(x)))
  , double(0()) -> 0()
  , square(s(x)) -> s(plus(square(x), double(x)))
  , square(0()) -> 0()
  , plus(n, s(m)) -> s(plus(n, m))
  , plus(n, 0()) -> n }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { log^#(x, s(s(y))) ->
    c_1(cond^#(le(x, s(s(y))), x, y), le^#(x, s(s(y))))
  , cond^#(true(), x, y) -> c_2()
  , cond^#(false(), x, y) ->
    c_3(double^#(log(x, square(s(s(y))))),
        log^#(x, square(s(s(y)))),
        square^#(s(s(y))))
  , le^#(s(u), s(v)) -> c_4(le^#(u, v))
  , le^#(s(u), 0()) -> c_5()
  , le^#(0(), v) -> c_6()
  , double^#(s(x)) -> c_7(double^#(x))
  , double^#(0()) -> c_8()
  , square^#(s(x)) ->
    c_9(plus^#(square(x), double(x)), square^#(x), double^#(x))
  , square^#(0()) -> c_10()
  , plus^#(n, s(m)) -> c_11(plus^#(n, m))
  , plus^#(n, 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:
  { log^#(x, s(s(y))) ->
    c_1(cond^#(le(x, s(s(y))), x, y), le^#(x, s(s(y))))
  , cond^#(true(), x, y) -> c_2()
  , cond^#(false(), x, y) ->
    c_3(double^#(log(x, square(s(s(y))))),
        log^#(x, square(s(s(y)))),
        square^#(s(s(y))))
  , le^#(s(u), s(v)) -> c_4(le^#(u, v))
  , le^#(s(u), 0()) -> c_5()
  , le^#(0(), v) -> c_6()
  , double^#(s(x)) -> c_7(double^#(x))
  , double^#(0()) -> c_8()
  , square^#(s(x)) ->
    c_9(plus^#(square(x), double(x)), square^#(x), double^#(x))
  , square^#(0()) -> c_10()
  , plus^#(n, s(m)) -> c_11(plus^#(n, m))
  , plus^#(n, 0()) -> c_12() }
Weak Trs:
  { log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
  , cond(true(), x, y) -> s(0())
  , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
  , le(s(u), s(v)) -> le(u, v)
  , le(s(u), 0()) -> false()
  , le(0(), v) -> true()
  , double(s(x)) -> s(s(double(x)))
  , double(0()) -> 0()
  , square(s(x)) -> s(plus(square(x), double(x)))
  , square(0()) -> 0()
  , plus(n, s(m)) -> s(plus(n, m))
  , plus(n, 0()) -> n }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: log^#(x, s(s(y))) ->
         c_1(cond^#(le(x, s(s(y))), x, y), le^#(x, s(s(y))))
    , 2: cond^#(true(), x, y) -> c_2()
    , 3: cond^#(false(), x, y) ->
         c_3(double^#(log(x, square(s(s(y))))),
             log^#(x, square(s(s(y)))),
             square^#(s(s(y))))
    , 4: le^#(s(u), s(v)) -> c_4(le^#(u, v))
    , 5: le^#(s(u), 0()) -> c_5()
    , 6: le^#(0(), v) -> c_6()
    , 7: double^#(s(x)) -> c_7(double^#(x))
    , 8: double^#(0()) -> c_8()
    , 9: square^#(s(x)) ->
         c_9(plus^#(square(x), double(x)), square^#(x), double^#(x))
    , 10: square^#(0()) -> c_10()
    , 11: plus^#(n, s(m)) -> c_11(plus^#(n, m))
    , 12: plus^#(n, 0()) -> c_12() }

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

Strict DPs:
  { log^#(x, s(s(y))) ->
    c_1(cond^#(le(x, s(s(y))), x, y), le^#(x, s(s(y))))
  , cond^#(false(), x, y) ->
    c_3(double^#(log(x, square(s(s(y))))),
        log^#(x, square(s(s(y)))),
        square^#(s(s(y))))
  , le^#(s(u), s(v)) -> c_4(le^#(u, v))
  , double^#(s(x)) -> c_7(double^#(x))
  , square^#(s(x)) ->
    c_9(plus^#(square(x), double(x)), square^#(x), double^#(x))
  , plus^#(n, s(m)) -> c_11(plus^#(n, m)) }
Weak DPs:
  { cond^#(true(), x, y) -> c_2()
  , le^#(s(u), 0()) -> c_5()
  , le^#(0(), v) -> c_6()
  , double^#(0()) -> c_8()
  , square^#(0()) -> c_10()
  , plus^#(n, 0()) -> c_12() }
Weak Trs:
  { log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
  , cond(true(), x, y) -> s(0())
  , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
  , le(s(u), s(v)) -> le(u, v)
  , le(s(u), 0()) -> false()
  , le(0(), v) -> true()
  , double(s(x)) -> s(s(double(x)))
  , double(0()) -> 0()
  , square(s(x)) -> s(plus(square(x), double(x)))
  , square(0()) -> 0()
  , plus(n, s(m)) -> s(plus(n, m))
  , plus(n, 0()) -> n }
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.

{ cond^#(true(), x, y) -> c_2()
, le^#(s(u), 0()) -> c_5()
, le^#(0(), v) -> c_6()
, double^#(0()) -> c_8()
, square^#(0()) -> c_10()
, plus^#(n, 0()) -> c_12() }

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

Strict DPs:
  { log^#(x, s(s(y))) ->
    c_1(cond^#(le(x, s(s(y))), x, y), le^#(x, s(s(y))))
  , cond^#(false(), x, y) ->
    c_3(double^#(log(x, square(s(s(y))))),
        log^#(x, square(s(s(y)))),
        square^#(s(s(y))))
  , le^#(s(u), s(v)) -> c_4(le^#(u, v))
  , double^#(s(x)) -> c_7(double^#(x))
  , square^#(s(x)) ->
    c_9(plus^#(square(x), double(x)), square^#(x), double^#(x))
  , plus^#(n, s(m)) -> c_11(plus^#(n, m)) }
Weak Trs:
  { log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
  , cond(true(), x, y) -> s(0())
  , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
  , le(s(u), s(v)) -> le(u, v)
  , le(s(u), 0()) -> false()
  , le(0(), v) -> true()
  , double(s(x)) -> s(s(double(x)))
  , double(0()) -> 0()
  , square(s(x)) -> s(plus(square(x), double(x)))
  , square(0()) -> 0()
  , plus(n, s(m)) -> s(plus(n, m))
  , plus(n, 0()) -> n }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..