MAYBE

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

Strict Trs:
  { f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
  , gt(s(u), s(v)) -> gt(u, v)
  , gt(s(u), 0()) -> true()
  , gt(0(), v) -> false()
  , trunc(s(s(x))) -> s(s(trunc(x)))
  , trunc(s(0())) -> 0()
  , trunc(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { f^#(true(), x, y) ->
    c_1(f^#(gt(x, y), trunc(x), s(y)), gt^#(x, y), trunc^#(x))
  , gt^#(s(u), s(v)) -> c_2(gt^#(u, v))
  , gt^#(s(u), 0()) -> c_3()
  , gt^#(0(), v) -> c_4()
  , trunc^#(s(s(x))) -> c_5(trunc^#(x))
  , trunc^#(s(0())) -> c_6()
  , trunc^#(0()) -> c_7() }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(true(), x, y) ->
    c_1(f^#(gt(x, y), trunc(x), s(y)), gt^#(x, y), trunc^#(x))
  , gt^#(s(u), s(v)) -> c_2(gt^#(u, v))
  , gt^#(s(u), 0()) -> c_3()
  , gt^#(0(), v) -> c_4()
  , trunc^#(s(s(x))) -> c_5(trunc^#(x))
  , trunc^#(s(0())) -> c_6()
  , trunc^#(0()) -> c_7() }
Weak Trs:
  { f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
  , gt(s(u), s(v)) -> gt(u, v)
  , gt(s(u), 0()) -> true()
  , gt(0(), v) -> false()
  , trunc(s(s(x))) -> s(s(trunc(x)))
  , trunc(s(0())) -> 0()
  , trunc(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {3,4,6,7} by applications
of Pre({3,4,6,7}) = {1,2,5}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(true(), x, y) ->
         c_1(f^#(gt(x, y), trunc(x), s(y)), gt^#(x, y), trunc^#(x))
    , 2: gt^#(s(u), s(v)) -> c_2(gt^#(u, v))
    , 3: gt^#(s(u), 0()) -> c_3()
    , 4: gt^#(0(), v) -> c_4()
    , 5: trunc^#(s(s(x))) -> c_5(trunc^#(x))
    , 6: trunc^#(s(0())) -> c_6()
    , 7: trunc^#(0()) -> c_7() }

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

Strict DPs:
  { f^#(true(), x, y) ->
    c_1(f^#(gt(x, y), trunc(x), s(y)), gt^#(x, y), trunc^#(x))
  , gt^#(s(u), s(v)) -> c_2(gt^#(u, v))
  , trunc^#(s(s(x))) -> c_5(trunc^#(x)) }
Weak DPs:
  { gt^#(s(u), 0()) -> c_3()
  , gt^#(0(), v) -> c_4()
  , trunc^#(s(0())) -> c_6()
  , trunc^#(0()) -> c_7() }
Weak Trs:
  { f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
  , gt(s(u), s(v)) -> gt(u, v)
  , gt(s(u), 0()) -> true()
  , gt(0(), v) -> false()
  , trunc(s(s(x))) -> s(s(trunc(x)))
  , trunc(s(0())) -> 0()
  , trunc(0()) -> 0() }
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.

{ gt^#(s(u), 0()) -> c_3()
, gt^#(0(), v) -> c_4()
, trunc^#(s(0())) -> c_6()
, trunc^#(0()) -> c_7() }

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

Strict DPs:
  { f^#(true(), x, y) ->
    c_1(f^#(gt(x, y), trunc(x), s(y)), gt^#(x, y), trunc^#(x))
  , gt^#(s(u), s(v)) -> c_2(gt^#(u, v))
  , trunc^#(s(s(x))) -> c_5(trunc^#(x)) }
Weak Trs:
  { f(true(), x, y) -> f(gt(x, y), trunc(x), s(y))
  , gt(s(u), s(v)) -> gt(u, v)
  , gt(s(u), 0()) -> true()
  , gt(0(), v) -> false()
  , trunc(s(s(x))) -> s(s(trunc(x)))
  , trunc(s(0())) -> 0()
  , trunc(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { gt(s(u), s(v)) -> gt(u, v)
    , gt(s(u), 0()) -> true()
    , gt(0(), v) -> false()
    , trunc(s(s(x))) -> s(s(trunc(x)))
    , trunc(s(0())) -> 0()
    , trunc(0()) -> 0() }

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

Strict DPs:
  { f^#(true(), x, y) ->
    c_1(f^#(gt(x, y), trunc(x), s(y)), gt^#(x, y), trunc^#(x))
  , gt^#(s(u), s(v)) -> c_2(gt^#(u, v))
  , trunc^#(s(s(x))) -> c_5(trunc^#(x)) }
Weak Trs:
  { gt(s(u), s(v)) -> gt(u, v)
  , gt(s(u), 0()) -> true()
  , gt(0(), v) -> false()
  , trunc(s(s(x))) -> s(s(trunc(x)))
  , trunc(s(0())) -> 0()
  , trunc(0()) -> 0() }
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..