MAYBE

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

Strict Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , log(x) -> log2(x, 0())
  , log2(x, y) -> if(le(x, s(0())), x, inc(y))
  , if(true(), x, s(y)) -> y
  , if(false(), x, y) -> log2(half(x), y) }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'Best' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)'
      failed due to the following reason:
      
      Computation stopped due to timeout after 30.0 seconds.
   
   2) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due
         to the following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
      2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the
         following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
   
   3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed
      due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   

2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { half^#(0()) -> c_1()
     , half^#(s(0())) -> c_2()
     , half^#(s(s(x))) -> c_3(half^#(x))
     , le^#(0(), y) -> c_4()
     , le^#(s(x), 0()) -> c_5()
     , le^#(s(x), s(y)) -> c_6(le^#(x, y))
     , inc^#(0()) -> c_7()
     , inc^#(s(x)) -> c_8(inc^#(x))
     , log^#(x) -> c_9(log2^#(x, 0()))
     , log2^#(x, y) -> c_10(if^#(le(x, s(0())), x, inc(y)))
     , if^#(true(), x, s(y)) -> c_11(y)
     , if^#(false(), x, y) -> c_12(log2^#(half(x), y)) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { half^#(0()) -> c_1()
     , half^#(s(0())) -> c_2()
     , half^#(s(s(x))) -> c_3(half^#(x))
     , le^#(0(), y) -> c_4()
     , le^#(s(x), 0()) -> c_5()
     , le^#(s(x), s(y)) -> c_6(le^#(x, y))
     , inc^#(0()) -> c_7()
     , inc^#(s(x)) -> c_8(inc^#(x))
     , log^#(x) -> c_9(log2^#(x, 0()))
     , log2^#(x, y) -> c_10(if^#(le(x, s(0())), x, inc(y)))
     , if^#(true(), x, s(y)) -> c_11(y)
     , if^#(false(), x, y) -> c_12(log2^#(half(x), y)) }
   Strict Trs:
     { half(0()) -> 0()
     , half(s(0())) -> 0()
     , half(s(s(x))) -> s(half(x))
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , inc(0()) -> 0()
     , inc(s(x)) -> s(inc(x))
     , log(x) -> log2(x, 0())
     , log2(x, y) -> if(le(x, s(0())), x, inc(y))
     , if(true(), x, s(y)) -> y
     , if(false(), x, y) -> log2(half(x), y) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,2,4,5,7} by
   applications of Pre({1,2,4,5,7}) = {3,6,8,11}. Here rules are
   labeled as follows:
   
     DPs:
       { 1: half^#(0()) -> c_1()
       , 2: half^#(s(0())) -> c_2()
       , 3: half^#(s(s(x))) -> c_3(half^#(x))
       , 4: le^#(0(), y) -> c_4()
       , 5: le^#(s(x), 0()) -> c_5()
       , 6: le^#(s(x), s(y)) -> c_6(le^#(x, y))
       , 7: inc^#(0()) -> c_7()
       , 8: inc^#(s(x)) -> c_8(inc^#(x))
       , 9: log^#(x) -> c_9(log2^#(x, 0()))
       , 10: log2^#(x, y) -> c_10(if^#(le(x, s(0())), x, inc(y)))
       , 11: if^#(true(), x, s(y)) -> c_11(y)
       , 12: if^#(false(), x, y) -> c_12(log2^#(half(x), y)) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { half^#(s(s(x))) -> c_3(half^#(x))
     , le^#(s(x), s(y)) -> c_6(le^#(x, y))
     , inc^#(s(x)) -> c_8(inc^#(x))
     , log^#(x) -> c_9(log2^#(x, 0()))
     , log2^#(x, y) -> c_10(if^#(le(x, s(0())), x, inc(y)))
     , if^#(true(), x, s(y)) -> c_11(y)
     , if^#(false(), x, y) -> c_12(log2^#(half(x), y)) }
   Strict Trs:
     { half(0()) -> 0()
     , half(s(0())) -> 0()
     , half(s(s(x))) -> s(half(x))
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , inc(0()) -> 0()
     , inc(s(x)) -> s(inc(x))
     , log(x) -> log2(x, 0())
     , log2(x, y) -> if(le(x, s(0())), x, inc(y))
     , if(true(), x, s(y)) -> y
     , if(false(), x, y) -> log2(half(x), y) }
   Weak DPs:
     { half^#(0()) -> c_1()
     , half^#(s(0())) -> c_2()
     , le^#(0(), y) -> c_4()
     , le^#(s(x), 0()) -> c_5()
     , inc^#(0()) -> c_7() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..