MAYBE

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

Strict Trs:
  { lt(x, 0()) -> false()
  , lt(0(), s(x)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , minus(x, y) -> help(lt(y, x), x, y)
  , help(true(), x, y) -> s(minus(x, s(y)))
  , help(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { lt^#(x, 0()) -> c_1()
  , lt^#(0(), s(x)) -> c_2()
  , lt^#(s(x), s(y)) -> c_3(lt^#(x, y))
  , minus^#(x, y) -> c_4(help^#(lt(y, x), x, y), lt^#(y, x))
  , help^#(true(), x, y) -> c_5(minus^#(x, s(y)))
  , help^#(false(), x, y) -> c_6() }

and mark the set of starting terms.

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

Strict DPs:
  { lt^#(x, 0()) -> c_1()
  , lt^#(0(), s(x)) -> c_2()
  , lt^#(s(x), s(y)) -> c_3(lt^#(x, y))
  , minus^#(x, y) -> c_4(help^#(lt(y, x), x, y), lt^#(y, x))
  , help^#(true(), x, y) -> c_5(minus^#(x, s(y)))
  , help^#(false(), x, y) -> c_6() }
Weak Trs:
  { lt(x, 0()) -> false()
  , lt(0(), s(x)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , minus(x, y) -> help(lt(y, x), x, y)
  , help(true(), x, y) -> s(minus(x, s(y)))
  , help(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: lt^#(x, 0()) -> c_1()
    , 2: lt^#(0(), s(x)) -> c_2()
    , 3: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))
    , 4: minus^#(x, y) -> c_4(help^#(lt(y, x), x, y), lt^#(y, x))
    , 5: help^#(true(), x, y) -> c_5(minus^#(x, s(y)))
    , 6: help^#(false(), x, y) -> c_6() }

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

Strict DPs:
  { lt^#(s(x), s(y)) -> c_3(lt^#(x, y))
  , minus^#(x, y) -> c_4(help^#(lt(y, x), x, y), lt^#(y, x))
  , help^#(true(), x, y) -> c_5(minus^#(x, s(y))) }
Weak DPs:
  { lt^#(x, 0()) -> c_1()
  , lt^#(0(), s(x)) -> c_2()
  , help^#(false(), x, y) -> c_6() }
Weak Trs:
  { lt(x, 0()) -> false()
  , lt(0(), s(x)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , minus(x, y) -> help(lt(y, x), x, y)
  , help(true(), x, y) -> s(minus(x, s(y)))
  , help(false(), x, y) -> 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.

{ lt^#(x, 0()) -> c_1()
, lt^#(0(), s(x)) -> c_2()
, help^#(false(), x, y) -> c_6() }

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

Strict DPs:
  { lt^#(s(x), s(y)) -> c_3(lt^#(x, y))
  , minus^#(x, y) -> c_4(help^#(lt(y, x), x, y), lt^#(y, x))
  , help^#(true(), x, y) -> c_5(minus^#(x, s(y))) }
Weak Trs:
  { lt(x, 0()) -> false()
  , lt(0(), s(x)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , minus(x, y) -> help(lt(y, x), x, y)
  , help(true(), x, y) -> s(minus(x, s(y)))
  , help(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { lt(x, 0()) -> false()
    , lt(0(), s(x)) -> true()
    , lt(s(x), s(y)) -> lt(x, y) }

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

Strict DPs:
  { lt^#(s(x), s(y)) -> c_3(lt^#(x, y))
  , minus^#(x, y) -> c_4(help^#(lt(y, x), x, y), lt^#(y, x))
  , help^#(true(), x, y) -> c_5(minus^#(x, s(y))) }
Weak Trs:
  { lt(x, 0()) -> false()
  , lt(0(), s(x)) -> true()
  , lt(s(x), s(y)) -> lt(x, y) }
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..