MAYBE

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

Strict Trs:
  { a(x1) -> b(c(x1))
  , a(b(x1)) -> b(a(x1))
  , a(c(x1)) -> c(a(x1))
  , d(c(x1)) -> d(a(x1)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a^#(x1) -> c_1()
  , a^#(b(x1)) -> c_2(a^#(x1))
  , a^#(c(x1)) -> c_3(a^#(x1))
  , d^#(c(x1)) -> c_4(d^#(a(x1)), a^#(x1)) }

and mark the set of starting terms.

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

Strict DPs:
  { a^#(x1) -> c_1()
  , a^#(b(x1)) -> c_2(a^#(x1))
  , a^#(c(x1)) -> c_3(a^#(x1))
  , d^#(c(x1)) -> c_4(d^#(a(x1)), a^#(x1)) }
Weak Trs:
  { a(x1) -> b(c(x1))
  , a(b(x1)) -> b(a(x1))
  , a(c(x1)) -> c(a(x1))
  , d(c(x1)) -> d(a(x1)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: a^#(x1) -> c_1()
    , 2: a^#(b(x1)) -> c_2(a^#(x1))
    , 3: a^#(c(x1)) -> c_3(a^#(x1))
    , 4: d^#(c(x1)) -> c_4(d^#(a(x1)), a^#(x1)) }

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

Strict DPs:
  { a^#(b(x1)) -> c_2(a^#(x1))
  , a^#(c(x1)) -> c_3(a^#(x1))
  , d^#(c(x1)) -> c_4(d^#(a(x1)), a^#(x1)) }
Weak DPs: { a^#(x1) -> c_1() }
Weak Trs:
  { a(x1) -> b(c(x1))
  , a(b(x1)) -> b(a(x1))
  , a(c(x1)) -> c(a(x1))
  , d(c(x1)) -> d(a(x1)) }
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.

{ a^#(x1) -> c_1() }

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

Strict DPs:
  { a^#(b(x1)) -> c_2(a^#(x1))
  , a^#(c(x1)) -> c_3(a^#(x1))
  , d^#(c(x1)) -> c_4(d^#(a(x1)), a^#(x1)) }
Weak Trs:
  { a(x1) -> b(c(x1))
  , a(b(x1)) -> b(a(x1))
  , a(c(x1)) -> c(a(x1))
  , d(c(x1)) -> d(a(x1)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { a(x1) -> b(c(x1))
    , a(b(x1)) -> b(a(x1))
    , a(c(x1)) -> c(a(x1)) }

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

Strict DPs:
  { a^#(b(x1)) -> c_2(a^#(x1))
  , a^#(c(x1)) -> c_3(a^#(x1))
  , d^#(c(x1)) -> c_4(d^#(a(x1)), a^#(x1)) }
Weak Trs:
  { a(x1) -> b(c(x1))
  , a(b(x1)) -> b(a(x1))
  , a(c(x1)) -> c(a(x1)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..