MAYBE

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

Strict Trs:
  { quot(x, 0(), s(z)) -> s(quot(x, plus(z, s(0())), s(z)))
  , quot(0(), s(y), s(z)) -> 0()
  , quot(s(x), s(y), z) -> quot(x, y, z)
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { quot^#(x, 0(), s(z)) ->
    c_1(quot^#(x, plus(z, s(0())), s(z)), plus^#(z, s(0())))
  , quot^#(0(), s(y), s(z)) -> c_2()
  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
  , plus^#(0(), y) -> c_4()
  , plus^#(s(x), y) -> c_5(plus^#(x, y)) }

and mark the set of starting terms.

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

Strict DPs:
  { quot^#(x, 0(), s(z)) ->
    c_1(quot^#(x, plus(z, s(0())), s(z)), plus^#(z, s(0())))
  , quot^#(0(), s(y), s(z)) -> c_2()
  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
  , plus^#(0(), y) -> c_4()
  , plus^#(s(x), y) -> c_5(plus^#(x, y)) }
Weak Trs:
  { quot(x, 0(), s(z)) -> s(quot(x, plus(z, s(0())), s(z)))
  , quot(0(), s(y), s(z)) -> 0()
  , quot(s(x), s(y), z) -> quot(x, y, z)
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: quot^#(x, 0(), s(z)) ->
         c_1(quot^#(x, plus(z, s(0())), s(z)), plus^#(z, s(0())))
    , 2: quot^#(0(), s(y), s(z)) -> c_2()
    , 3: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
    , 4: plus^#(0(), y) -> c_4()
    , 5: plus^#(s(x), y) -> c_5(plus^#(x, y)) }

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

Strict DPs:
  { quot^#(x, 0(), s(z)) ->
    c_1(quot^#(x, plus(z, s(0())), s(z)), plus^#(z, s(0())))
  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
  , plus^#(s(x), y) -> c_5(plus^#(x, y)) }
Weak DPs:
  { quot^#(0(), s(y), s(z)) -> c_2()
  , plus^#(0(), y) -> c_4() }
Weak Trs:
  { quot(x, 0(), s(z)) -> s(quot(x, plus(z, s(0())), s(z)))
  , quot(0(), s(y), s(z)) -> 0()
  , quot(s(x), s(y), z) -> quot(x, y, z)
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(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.

{ quot^#(0(), s(y), s(z)) -> c_2()
, plus^#(0(), y) -> c_4() }

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

Strict DPs:
  { quot^#(x, 0(), s(z)) ->
    c_1(quot^#(x, plus(z, s(0())), s(z)), plus^#(z, s(0())))
  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
  , plus^#(s(x), y) -> c_5(plus^#(x, y)) }
Weak Trs:
  { quot(x, 0(), s(z)) -> s(quot(x, plus(z, s(0())), s(z)))
  , quot(0(), s(y), s(z)) -> 0()
  , quot(s(x), s(y), z) -> quot(x, y, z)
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { plus(0(), y) -> y
    , plus(s(x), y) -> s(plus(x, y)) }

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

Strict DPs:
  { quot^#(x, 0(), s(z)) ->
    c_1(quot^#(x, plus(z, s(0())), s(z)), plus^#(z, s(0())))
  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
  , plus^#(s(x), y) -> c_5(plus^#(x, y)) }
Weak Trs:
  { plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..