MAYBE

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

Strict Trs:
  { plus(x, y) -> if(isZero(x), y, s(plus(p(x), y)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , isZero(s(x)) -> false()
  , isZero(0()) -> true()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { plus^#(x, y) ->
    c_1(if^#(isZero(x), y, s(plus(p(x), y))),
        isZero^#(x),
        plus^#(p(x), y),
        p^#(x))
  , if^#(true(), x, y) -> c_2()
  , if^#(false(), x, y) -> c_3()
  , isZero^#(s(x)) -> c_4()
  , isZero^#(0()) -> c_5()
  , p^#(s(x)) -> c_6() }

and mark the set of starting terms.

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

Strict DPs:
  { plus^#(x, y) ->
    c_1(if^#(isZero(x), y, s(plus(p(x), y))),
        isZero^#(x),
        plus^#(p(x), y),
        p^#(x))
  , if^#(true(), x, y) -> c_2()
  , if^#(false(), x, y) -> c_3()
  , isZero^#(s(x)) -> c_4()
  , isZero^#(0()) -> c_5()
  , p^#(s(x)) -> c_6() }
Weak Trs:
  { plus(x, y) -> if(isZero(x), y, s(plus(p(x), y)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , isZero(s(x)) -> false()
  , isZero(0()) -> true()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: plus^#(x, y) ->
         c_1(if^#(isZero(x), y, s(plus(p(x), y))),
             isZero^#(x),
             plus^#(p(x), y),
             p^#(x))
    , 2: if^#(true(), x, y) -> c_2()
    , 3: if^#(false(), x, y) -> c_3()
    , 4: isZero^#(s(x)) -> c_4()
    , 5: isZero^#(0()) -> c_5()
    , 6: p^#(s(x)) -> c_6() }

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

Strict DPs:
  { plus^#(x, y) ->
    c_1(if^#(isZero(x), y, s(plus(p(x), y))),
        isZero^#(x),
        plus^#(p(x), y),
        p^#(x)) }
Weak DPs:
  { if^#(true(), x, y) -> c_2()
  , if^#(false(), x, y) -> c_3()
  , isZero^#(s(x)) -> c_4()
  , isZero^#(0()) -> c_5()
  , p^#(s(x)) -> c_6() }
Weak Trs:
  { plus(x, y) -> if(isZero(x), y, s(plus(p(x), y)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , isZero(s(x)) -> false()
  , isZero(0()) -> true()
  , 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.

{ if^#(true(), x, y) -> c_2()
, if^#(false(), x, y) -> c_3()
, isZero^#(s(x)) -> c_4()
, isZero^#(0()) -> c_5()
, p^#(s(x)) -> c_6() }

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

Strict DPs:
  { plus^#(x, y) ->
    c_1(if^#(isZero(x), y, s(plus(p(x), y))),
        isZero^#(x),
        plus^#(p(x), y),
        p^#(x)) }
Weak Trs:
  { plus(x, y) -> if(isZero(x), y, s(plus(p(x), y)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , isZero(s(x)) -> false()
  , isZero(0()) -> true()
  , 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:

  { plus^#(x, y) ->
    c_1(if^#(isZero(x), y, s(plus(p(x), y))),
        isZero^#(x),
        plus^#(p(x), y),
        p^#(x)) }

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

Strict DPs: { plus^#(x, y) -> c_1(plus^#(p(x), y)) }
Weak Trs:
  { plus(x, y) -> if(isZero(x), y, s(plus(p(x), y)))
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , isZero(s(x)) -> false()
  , isZero(0()) -> true()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules: { p(s(x)) -> x }

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

Strict DPs: { plus^#(x, y) -> c_1(plus^#(p(x), y)) }
Weak Trs: { p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..