MAYBE

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

Strict Trs:
  { p(0()) -> 0()
  , p(s(X)) -> X
  , leq(0(), Y) -> true()
  , leq(s(X), 0()) -> false()
  , leq(s(X), s(Y)) -> leq(X, Y)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> Y
  , diff(X, Y) -> if(leq(X, Y), 0(), s(diff(p(X), Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { p^#(0()) -> c_1()
  , p^#(s(X)) -> c_2()
  , leq^#(0(), Y) -> c_3()
  , leq^#(s(X), 0()) -> c_4()
  , leq^#(s(X), s(Y)) -> c_5(leq^#(X, Y))
  , if^#(true(), X, Y) -> c_6()
  , if^#(false(), X, Y) -> c_7()
  , diff^#(X, Y) ->
    c_8(if^#(leq(X, Y), 0(), s(diff(p(X), Y))),
        leq^#(X, Y),
        diff^#(p(X), Y),
        p^#(X)) }

and mark the set of starting terms.

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

Strict DPs:
  { p^#(0()) -> c_1()
  , p^#(s(X)) -> c_2()
  , leq^#(0(), Y) -> c_3()
  , leq^#(s(X), 0()) -> c_4()
  , leq^#(s(X), s(Y)) -> c_5(leq^#(X, Y))
  , if^#(true(), X, Y) -> c_6()
  , if^#(false(), X, Y) -> c_7()
  , diff^#(X, Y) ->
    c_8(if^#(leq(X, Y), 0(), s(diff(p(X), Y))),
        leq^#(X, Y),
        diff^#(p(X), Y),
        p^#(X)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(X)) -> X
  , leq(0(), Y) -> true()
  , leq(s(X), 0()) -> false()
  , leq(s(X), s(Y)) -> leq(X, Y)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> Y
  , diff(X, Y) -> if(leq(X, Y), 0(), s(diff(p(X), Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,4,6,7} by
applications of Pre({1,2,3,4,6,7}) = {5,8}. Here rules are labeled
as follows:

  DPs:
    { 1: p^#(0()) -> c_1()
    , 2: p^#(s(X)) -> c_2()
    , 3: leq^#(0(), Y) -> c_3()
    , 4: leq^#(s(X), 0()) -> c_4()
    , 5: leq^#(s(X), s(Y)) -> c_5(leq^#(X, Y))
    , 6: if^#(true(), X, Y) -> c_6()
    , 7: if^#(false(), X, Y) -> c_7()
    , 8: diff^#(X, Y) ->
         c_8(if^#(leq(X, Y), 0(), s(diff(p(X), Y))),
             leq^#(X, Y),
             diff^#(p(X), Y),
             p^#(X)) }

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

Strict DPs:
  { leq^#(s(X), s(Y)) -> c_5(leq^#(X, Y))
  , diff^#(X, Y) ->
    c_8(if^#(leq(X, Y), 0(), s(diff(p(X), Y))),
        leq^#(X, Y),
        diff^#(p(X), Y),
        p^#(X)) }
Weak DPs:
  { p^#(0()) -> c_1()
  , p^#(s(X)) -> c_2()
  , leq^#(0(), Y) -> c_3()
  , leq^#(s(X), 0()) -> c_4()
  , if^#(true(), X, Y) -> c_6()
  , if^#(false(), X, Y) -> c_7() }
Weak Trs:
  { p(0()) -> 0()
  , p(s(X)) -> X
  , leq(0(), Y) -> true()
  , leq(s(X), 0()) -> false()
  , leq(s(X), s(Y)) -> leq(X, Y)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> Y
  , diff(X, Y) -> if(leq(X, Y), 0(), s(diff(p(X), Y))) }
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.

{ p^#(0()) -> c_1()
, p^#(s(X)) -> c_2()
, leq^#(0(), Y) -> c_3()
, leq^#(s(X), 0()) -> c_4()
, if^#(true(), X, Y) -> c_6()
, if^#(false(), X, Y) -> c_7() }

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

Strict DPs:
  { leq^#(s(X), s(Y)) -> c_5(leq^#(X, Y))
  , diff^#(X, Y) ->
    c_8(if^#(leq(X, Y), 0(), s(diff(p(X), Y))),
        leq^#(X, Y),
        diff^#(p(X), Y),
        p^#(X)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(X)) -> X
  , leq(0(), Y) -> true()
  , leq(s(X), 0()) -> false()
  , leq(s(X), s(Y)) -> leq(X, Y)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> Y
  , diff(X, Y) -> if(leq(X, Y), 0(), s(diff(p(X), Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { diff^#(X, Y) ->
    c_8(if^#(leq(X, Y), 0(), s(diff(p(X), Y))),
        leq^#(X, Y),
        diff^#(p(X), Y),
        p^#(X)) }

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

Strict DPs:
  { leq^#(s(X), s(Y)) -> c_1(leq^#(X, Y))
  , diff^#(X, Y) -> c_2(leq^#(X, Y), diff^#(p(X), Y)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(X)) -> X
  , leq(0(), Y) -> true()
  , leq(s(X), 0()) -> false()
  , leq(s(X), s(Y)) -> leq(X, Y)
  , if(true(), X, Y) -> X
  , if(false(), X, Y) -> Y
  , diff(X, Y) -> if(leq(X, Y), 0(), s(diff(p(X), Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { p(0()) -> 0()
    , p(s(X)) -> X }

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

Strict DPs:
  { leq^#(s(X), s(Y)) -> c_1(leq^#(X, Y))
  , diff^#(X, Y) -> c_2(leq^#(X, Y), diff^#(p(X), Y)) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(X)) -> X }
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..