MAYBE

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

Strict Trs:
  { f(tt(), x) -> f(isNat(x), s(x))
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> tt() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { f^#(tt(), x) -> c_1(f^#(isNat(x), s(x)), isNat^#(x))
  , isNat^#(s(x)) -> c_2(isNat^#(x))
  , isNat^#(0()) -> c_3() }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(tt(), x) -> c_1(f^#(isNat(x), s(x)), isNat^#(x))
  , isNat^#(s(x)) -> c_2(isNat^#(x))
  , isNat^#(0()) -> c_3() }
Weak Trs:
  { f(tt(), x) -> f(isNat(x), s(x))
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> tt() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: f^#(tt(), x) -> c_1(f^#(isNat(x), s(x)), isNat^#(x))
    , 2: isNat^#(s(x)) -> c_2(isNat^#(x))
    , 3: isNat^#(0()) -> c_3() }

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

Strict DPs:
  { f^#(tt(), x) -> c_1(f^#(isNat(x), s(x)), isNat^#(x))
  , isNat^#(s(x)) -> c_2(isNat^#(x)) }
Weak DPs: { isNat^#(0()) -> c_3() }
Weak Trs:
  { f(tt(), x) -> f(isNat(x), s(x))
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> tt() }
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.

{ isNat^#(0()) -> c_3() }

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

Strict DPs:
  { f^#(tt(), x) -> c_1(f^#(isNat(x), s(x)), isNat^#(x))
  , isNat^#(s(x)) -> c_2(isNat^#(x)) }
Weak Trs:
  { f(tt(), x) -> f(isNat(x), s(x))
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> tt() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { isNat(s(x)) -> isNat(x)
    , isNat(0()) -> tt() }

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

Strict DPs:
  { f^#(tt(), x) -> c_1(f^#(isNat(x), s(x)), isNat^#(x))
  , isNat^#(s(x)) -> c_2(isNat^#(x)) }
Weak Trs:
  { isNat(s(x)) -> isNat(x)
  , isNat(0()) -> tt() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..