MAYBE

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

Strict Trs:
  { +(x, 0()) -> x
  , +(+(x, y), z) -> +(x, +(y, z))
  , +(0(), x) -> x
  , +(s(x), s(y)) -> s(s(+(x, y)))
  , *(x, 0()) -> 0()
  , *(0(), x) -> 0()
  , *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
  , *(*(x, y), z) -> *(x, *(y, z))
  , sum(nil()) -> 0()
  , sum(cons(x, l)) -> +(x, sum(l))
  , prod(nil()) -> s(0())
  , prod(cons(x, l)) -> *(x, prod(l)) }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
   following reason:
   
   Computation stopped due to timeout after 60.0 seconds.

2) '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) '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.
      
   
   3) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) '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
      
      2) '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
      
   

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


Arrrr..