MAYBE

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

Strict Trs:
  { 0(#()) -> #()
  , +(x, #()) -> x
  , +(0(x), 0(y)) -> 0(+(x, y))
  , +(0(x), 1(y)) -> 1(+(x, y))
  , +(#(), x) -> x
  , +(+(x, y), z) -> +(x, +(y, z))
  , +(1(x), 0(y)) -> 1(+(x, y))
  , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#())))
  , *(x, +(y, z)) -> +(*(x, y), *(x, z))
  , *(0(x), y) -> 0(*(x, y))
  , *(#(), x) -> #()
  , *(1(x), y) -> +(0(*(x, y)), y)
  , *(*(x, y), z) -> *(x, *(y, z))
  , app(nil(), l) -> l
  , app(cons(x, l1), l2) -> cons(x, app(l1, l2))
  , sum(app(l1, l2)) -> +(sum(l1), sum(l2))
  , sum(nil()) -> 0(#())
  , sum(cons(x, l)) -> +(x, sum(l))
  , prod(app(l1, l2)) -> *(prod(l1), prod(l2))
  , prod(nil()) -> 1(#())
  , 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) '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) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { 0^#(#()) -> c_1()
     , +^#(x, #()) -> c_2(x)
     , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
     , +^#(0(x), 1(y)) -> c_4(+^#(x, y))
     , +^#(#(), x) -> c_5(x)
     , +^#(+(x, y), z) -> c_6(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_7(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_8(0^#(+(+(x, y), 1(#()))))
     , *^#(x, +(y, z)) -> c_9(+^#(*(x, y), *(x, z)))
     , *^#(0(x), y) -> c_10(0^#(*(x, y)))
     , *^#(#(), x) -> c_11()
     , *^#(1(x), y) -> c_12(+^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_13(*^#(x, *(y, z)))
     , app^#(nil(), l) -> c_14(l)
     , app^#(cons(x, l1), l2) -> c_15(x, app^#(l1, l2))
     , sum^#(app(l1, l2)) -> c_16(+^#(sum(l1), sum(l2)))
     , sum^#(nil()) -> c_17(0^#(#()))
     , sum^#(cons(x, l)) -> c_18(+^#(x, sum(l)))
     , prod^#(app(l1, l2)) -> c_19(*^#(prod(l1), prod(l2)))
     , prod^#(nil()) -> c_20()
     , prod^#(cons(x, l)) -> c_21(*^#(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:
     { 0^#(#()) -> c_1()
     , +^#(x, #()) -> c_2(x)
     , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
     , +^#(0(x), 1(y)) -> c_4(+^#(x, y))
     , +^#(#(), x) -> c_5(x)
     , +^#(+(x, y), z) -> c_6(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_7(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_8(0^#(+(+(x, y), 1(#()))))
     , *^#(x, +(y, z)) -> c_9(+^#(*(x, y), *(x, z)))
     , *^#(0(x), y) -> c_10(0^#(*(x, y)))
     , *^#(#(), x) -> c_11()
     , *^#(1(x), y) -> c_12(+^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_13(*^#(x, *(y, z)))
     , app^#(nil(), l) -> c_14(l)
     , app^#(cons(x, l1), l2) -> c_15(x, app^#(l1, l2))
     , sum^#(app(l1, l2)) -> c_16(+^#(sum(l1), sum(l2)))
     , sum^#(nil()) -> c_17(0^#(#()))
     , sum^#(cons(x, l)) -> c_18(+^#(x, sum(l)))
     , prod^#(app(l1, l2)) -> c_19(*^#(prod(l1), prod(l2)))
     , prod^#(nil()) -> c_20()
     , prod^#(cons(x, l)) -> c_21(*^#(x, prod(l))) }
   Strict Trs:
     { 0(#()) -> #()
     , +(x, #()) -> x
     , +(0(x), 0(y)) -> 0(+(x, y))
     , +(0(x), 1(y)) -> 1(+(x, y))
     , +(#(), x) -> x
     , +(+(x, y), z) -> +(x, +(y, z))
     , +(1(x), 0(y)) -> 1(+(x, y))
     , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#())))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(0(x), y) -> 0(*(x, y))
     , *(#(), x) -> #()
     , *(1(x), y) -> +(0(*(x, y)), y)
     , *(*(x, y), z) -> *(x, *(y, z))
     , app(nil(), l) -> l
     , app(cons(x, l1), l2) -> cons(x, app(l1, l2))
     , sum(app(l1, l2)) -> +(sum(l1), sum(l2))
     , sum(nil()) -> 0(#())
     , sum(cons(x, l)) -> +(x, sum(l))
     , prod(app(l1, l2)) -> *(prod(l1), prod(l2))
     , prod(nil()) -> 1(#())
     , prod(cons(x, l)) -> *(x, prod(l)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,11,20} by applications
   of Pre({1,11,20}) = {2,3,5,8,10,13,14,15,17,19,21}. Here rules are
   labeled as follows:
   
     DPs:
       { 1: 0^#(#()) -> c_1()
       , 2: +^#(x, #()) -> c_2(x)
       , 3: +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
       , 4: +^#(0(x), 1(y)) -> c_4(+^#(x, y))
       , 5: +^#(#(), x) -> c_5(x)
       , 6: +^#(+(x, y), z) -> c_6(+^#(x, +(y, z)))
       , 7: +^#(1(x), 0(y)) -> c_7(+^#(x, y))
       , 8: +^#(1(x), 1(y)) -> c_8(0^#(+(+(x, y), 1(#()))))
       , 9: *^#(x, +(y, z)) -> c_9(+^#(*(x, y), *(x, z)))
       , 10: *^#(0(x), y) -> c_10(0^#(*(x, y)))
       , 11: *^#(#(), x) -> c_11()
       , 12: *^#(1(x), y) -> c_12(+^#(0(*(x, y)), y))
       , 13: *^#(*(x, y), z) -> c_13(*^#(x, *(y, z)))
       , 14: app^#(nil(), l) -> c_14(l)
       , 15: app^#(cons(x, l1), l2) -> c_15(x, app^#(l1, l2))
       , 16: sum^#(app(l1, l2)) -> c_16(+^#(sum(l1), sum(l2)))
       , 17: sum^#(nil()) -> c_17(0^#(#()))
       , 18: sum^#(cons(x, l)) -> c_18(+^#(x, sum(l)))
       , 19: prod^#(app(l1, l2)) -> c_19(*^#(prod(l1), prod(l2)))
       , 20: prod^#(nil()) -> c_20()
       , 21: prod^#(cons(x, l)) -> c_21(*^#(x, prod(l))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { +^#(x, #()) -> c_2(x)
     , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
     , +^#(0(x), 1(y)) -> c_4(+^#(x, y))
     , +^#(#(), x) -> c_5(x)
     , +^#(+(x, y), z) -> c_6(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_7(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_8(0^#(+(+(x, y), 1(#()))))
     , *^#(x, +(y, z)) -> c_9(+^#(*(x, y), *(x, z)))
     , *^#(0(x), y) -> c_10(0^#(*(x, y)))
     , *^#(1(x), y) -> c_12(+^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_13(*^#(x, *(y, z)))
     , app^#(nil(), l) -> c_14(l)
     , app^#(cons(x, l1), l2) -> c_15(x, app^#(l1, l2))
     , sum^#(app(l1, l2)) -> c_16(+^#(sum(l1), sum(l2)))
     , sum^#(nil()) -> c_17(0^#(#()))
     , sum^#(cons(x, l)) -> c_18(+^#(x, sum(l)))
     , prod^#(app(l1, l2)) -> c_19(*^#(prod(l1), prod(l2)))
     , prod^#(cons(x, l)) -> c_21(*^#(x, prod(l))) }
   Strict Trs:
     { 0(#()) -> #()
     , +(x, #()) -> x
     , +(0(x), 0(y)) -> 0(+(x, y))
     , +(0(x), 1(y)) -> 1(+(x, y))
     , +(#(), x) -> x
     , +(+(x, y), z) -> +(x, +(y, z))
     , +(1(x), 0(y)) -> 1(+(x, y))
     , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#())))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(0(x), y) -> 0(*(x, y))
     , *(#(), x) -> #()
     , *(1(x), y) -> +(0(*(x, y)), y)
     , *(*(x, y), z) -> *(x, *(y, z))
     , app(nil(), l) -> l
     , app(cons(x, l1), l2) -> cons(x, app(l1, l2))
     , sum(app(l1, l2)) -> +(sum(l1), sum(l2))
     , sum(nil()) -> 0(#())
     , sum(cons(x, l)) -> +(x, sum(l))
     , prod(app(l1, l2)) -> *(prod(l1), prod(l2))
     , prod(nil()) -> 1(#())
     , prod(cons(x, l)) -> *(x, prod(l)) }
   Weak DPs:
     { 0^#(#()) -> c_1()
     , *^#(#(), x) -> c_11()
     , prod^#(nil()) -> c_20() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,7,9,15} by applications
   of Pre({2,7,9,15}) = {1,3,4,5,6,8,10,11,12,13,14,16,17,18}. Here
   rules are labeled as follows:
   
     DPs:
       { 1: +^#(x, #()) -> c_2(x)
       , 2: +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
       , 3: +^#(0(x), 1(y)) -> c_4(+^#(x, y))
       , 4: +^#(#(), x) -> c_5(x)
       , 5: +^#(+(x, y), z) -> c_6(+^#(x, +(y, z)))
       , 6: +^#(1(x), 0(y)) -> c_7(+^#(x, y))
       , 7: +^#(1(x), 1(y)) -> c_8(0^#(+(+(x, y), 1(#()))))
       , 8: *^#(x, +(y, z)) -> c_9(+^#(*(x, y), *(x, z)))
       , 9: *^#(0(x), y) -> c_10(0^#(*(x, y)))
       , 10: *^#(1(x), y) -> c_12(+^#(0(*(x, y)), y))
       , 11: *^#(*(x, y), z) -> c_13(*^#(x, *(y, z)))
       , 12: app^#(nil(), l) -> c_14(l)
       , 13: app^#(cons(x, l1), l2) -> c_15(x, app^#(l1, l2))
       , 14: sum^#(app(l1, l2)) -> c_16(+^#(sum(l1), sum(l2)))
       , 15: sum^#(nil()) -> c_17(0^#(#()))
       , 16: sum^#(cons(x, l)) -> c_18(+^#(x, sum(l)))
       , 17: prod^#(app(l1, l2)) -> c_19(*^#(prod(l1), prod(l2)))
       , 18: prod^#(cons(x, l)) -> c_21(*^#(x, prod(l)))
       , 19: 0^#(#()) -> c_1()
       , 20: *^#(#(), x) -> c_11()
       , 21: prod^#(nil()) -> c_20() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { +^#(x, #()) -> c_2(x)
     , +^#(0(x), 1(y)) -> c_4(+^#(x, y))
     , +^#(#(), x) -> c_5(x)
     , +^#(+(x, y), z) -> c_6(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_7(+^#(x, y))
     , *^#(x, +(y, z)) -> c_9(+^#(*(x, y), *(x, z)))
     , *^#(1(x), y) -> c_12(+^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_13(*^#(x, *(y, z)))
     , app^#(nil(), l) -> c_14(l)
     , app^#(cons(x, l1), l2) -> c_15(x, app^#(l1, l2))
     , sum^#(app(l1, l2)) -> c_16(+^#(sum(l1), sum(l2)))
     , sum^#(cons(x, l)) -> c_18(+^#(x, sum(l)))
     , prod^#(app(l1, l2)) -> c_19(*^#(prod(l1), prod(l2)))
     , prod^#(cons(x, l)) -> c_21(*^#(x, prod(l))) }
   Strict Trs:
     { 0(#()) -> #()
     , +(x, #()) -> x
     , +(0(x), 0(y)) -> 0(+(x, y))
     , +(0(x), 1(y)) -> 1(+(x, y))
     , +(#(), x) -> x
     , +(+(x, y), z) -> +(x, +(y, z))
     , +(1(x), 0(y)) -> 1(+(x, y))
     , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#())))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(0(x), y) -> 0(*(x, y))
     , *(#(), x) -> #()
     , *(1(x), y) -> +(0(*(x, y)), y)
     , *(*(x, y), z) -> *(x, *(y, z))
     , app(nil(), l) -> l
     , app(cons(x, l1), l2) -> cons(x, app(l1, l2))
     , sum(app(l1, l2)) -> +(sum(l1), sum(l2))
     , sum(nil()) -> 0(#())
     , sum(cons(x, l)) -> +(x, sum(l))
     , prod(app(l1, l2)) -> *(prod(l1), prod(l2))
     , prod(nil()) -> 1(#())
     , prod(cons(x, l)) -> *(x, prod(l)) }
   Weak DPs:
     { 0^#(#()) -> c_1()
     , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
     , +^#(1(x), 1(y)) -> c_8(0^#(+(+(x, y), 1(#()))))
     , *^#(0(x), y) -> c_10(0^#(*(x, y)))
     , *^#(#(), x) -> c_11()
     , sum^#(nil()) -> c_17(0^#(#()))
     , prod^#(nil()) -> c_20() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..