MAYBE

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

Strict Trs:
  { nats() -> cons(0(), incr(nats()))
  , incr(cons(X, XS)) -> cons(s(X), incr(XS))
  , pairs() -> cons(0(), incr(odds()))
  , odds() -> incr(pairs())
  , head(cons(X, XS)) -> X
  , tail(cons(X, XS)) -> XS }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { nats^#() -> c_1(incr^#(nats()), nats^#())
  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
  , pairs^#() -> c_3(incr^#(odds()), odds^#())
  , odds^#() -> c_4(incr^#(pairs()), pairs^#())
  , head^#(cons(X, XS)) -> c_5()
  , tail^#(cons(X, XS)) -> c_6() }

and mark the set of starting terms.

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

Strict DPs:
  { nats^#() -> c_1(incr^#(nats()), nats^#())
  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
  , pairs^#() -> c_3(incr^#(odds()), odds^#())
  , odds^#() -> c_4(incr^#(pairs()), pairs^#())
  , head^#(cons(X, XS)) -> c_5()
  , tail^#(cons(X, XS)) -> c_6() }
Weak Trs:
  { nats() -> cons(0(), incr(nats()))
  , incr(cons(X, XS)) -> cons(s(X), incr(XS))
  , pairs() -> cons(0(), incr(odds()))
  , odds() -> incr(pairs())
  , head(cons(X, XS)) -> X
  , tail(cons(X, XS)) -> XS }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: nats^#() -> c_1(incr^#(nats()), nats^#())
    , 2: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
    , 3: pairs^#() -> c_3(incr^#(odds()), odds^#())
    , 4: odds^#() -> c_4(incr^#(pairs()), pairs^#())
    , 5: head^#(cons(X, XS)) -> c_5()
    , 6: tail^#(cons(X, XS)) -> c_6() }

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

Strict DPs:
  { nats^#() -> c_1(incr^#(nats()), nats^#())
  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
  , pairs^#() -> c_3(incr^#(odds()), odds^#())
  , odds^#() -> c_4(incr^#(pairs()), pairs^#()) }
Weak DPs:
  { head^#(cons(X, XS)) -> c_5()
  , tail^#(cons(X, XS)) -> c_6() }
Weak Trs:
  { nats() -> cons(0(), incr(nats()))
  , incr(cons(X, XS)) -> cons(s(X), incr(XS))
  , pairs() -> cons(0(), incr(odds()))
  , odds() -> incr(pairs())
  , head(cons(X, XS)) -> X
  , tail(cons(X, XS)) -> XS }
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.

{ head^#(cons(X, XS)) -> c_5()
, tail^#(cons(X, XS)) -> c_6() }

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

Strict DPs:
  { nats^#() -> c_1(incr^#(nats()), nats^#())
  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
  , pairs^#() -> c_3(incr^#(odds()), odds^#())
  , odds^#() -> c_4(incr^#(pairs()), pairs^#()) }
Weak Trs:
  { nats() -> cons(0(), incr(nats()))
  , incr(cons(X, XS)) -> cons(s(X), incr(XS))
  , pairs() -> cons(0(), incr(odds()))
  , odds() -> incr(pairs())
  , head(cons(X, XS)) -> X
  , tail(cons(X, XS)) -> XS }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { nats() -> cons(0(), incr(nats()))
    , incr(cons(X, XS)) -> cons(s(X), incr(XS))
    , pairs() -> cons(0(), incr(odds()))
    , odds() -> incr(pairs()) }

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

Strict DPs:
  { nats^#() -> c_1(incr^#(nats()), nats^#())
  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
  , pairs^#() -> c_3(incr^#(odds()), odds^#())
  , odds^#() -> c_4(incr^#(pairs()), pairs^#()) }
Weak Trs:
  { nats() -> cons(0(), incr(nats()))
  , incr(cons(X, XS)) -> cons(s(X), incr(XS))
  , pairs() -> cons(0(), incr(odds()))
  , odds() -> incr(pairs()) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..