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))
  , +(0(x), j(y)) -> j(+(x, y))
  , +(#(), x) -> x
  , +(+(x, y), z) -> +(x, +(y, z))
  , +(1(x), 0(y)) -> 1(+(x, y))
  , +(1(x), 1(y)) -> j(+(+(x, y), 1(#())))
  , +(1(x), j(y)) -> 0(+(x, y))
  , +(j(x), 0(y)) -> j(+(x, y))
  , +(j(x), 1(y)) -> 0(+(x, y))
  , +(j(x), j(y)) -> 1(+(+(x, y), j(#())))
  , opp(0(x)) -> 0(opp(x))
  , opp(#()) -> #()
  , opp(1(x)) -> j(opp(x))
  , opp(j(x)) -> 1(opp(x))
  , -(x, y) -> +(x, opp(y))
  , *(x, +(y, z)) -> +(*(x, y), *(x, z))
  , *(0(x), y) -> 0(*(x, y))
  , *(#(), x) -> #()
  , *(+(x, y), z) -> +(*(x, z), *(y, z))
  , *(1(x), y) -> +(0(*(x, y)), y)
  , *(j(x), y) -> -(0(*(x, y)), y)
  , *(*(x, y), z) -> *(x, *(y, z)) }
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) '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.
      
   

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))
     , +^#(0(x), j(y)) -> c_5(+^#(x, y))
     , +^#(#(), x) -> c_6(x)
     , +^#(+(x, y), z) -> c_7(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_8(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_9(+^#(+(x, y), 1(#())))
     , +^#(1(x), j(y)) -> c_10(0^#(+(x, y)))
     , +^#(j(x), 0(y)) -> c_11(+^#(x, y))
     , +^#(j(x), 1(y)) -> c_12(0^#(+(x, y)))
     , +^#(j(x), j(y)) -> c_13(+^#(+(x, y), j(#())))
     , opp^#(0(x)) -> c_14(0^#(opp(x)))
     , opp^#(#()) -> c_15()
     , opp^#(1(x)) -> c_16(opp^#(x))
     , opp^#(j(x)) -> c_17(opp^#(x))
     , -^#(x, y) -> c_18(+^#(x, opp(y)))
     , *^#(x, +(y, z)) -> c_19(+^#(*(x, y), *(x, z)))
     , *^#(0(x), y) -> c_20(0^#(*(x, y)))
     , *^#(#(), x) -> c_21()
     , *^#(+(x, y), z) -> c_22(+^#(*(x, z), *(y, z)))
     , *^#(1(x), y) -> c_23(+^#(0(*(x, y)), y))
     , *^#(j(x), y) -> c_24(-^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_25(*^#(x, *(y, z))) }
   
   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))
     , +^#(0(x), j(y)) -> c_5(+^#(x, y))
     , +^#(#(), x) -> c_6(x)
     , +^#(+(x, y), z) -> c_7(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_8(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_9(+^#(+(x, y), 1(#())))
     , +^#(1(x), j(y)) -> c_10(0^#(+(x, y)))
     , +^#(j(x), 0(y)) -> c_11(+^#(x, y))
     , +^#(j(x), 1(y)) -> c_12(0^#(+(x, y)))
     , +^#(j(x), j(y)) -> c_13(+^#(+(x, y), j(#())))
     , opp^#(0(x)) -> c_14(0^#(opp(x)))
     , opp^#(#()) -> c_15()
     , opp^#(1(x)) -> c_16(opp^#(x))
     , opp^#(j(x)) -> c_17(opp^#(x))
     , -^#(x, y) -> c_18(+^#(x, opp(y)))
     , *^#(x, +(y, z)) -> c_19(+^#(*(x, y), *(x, z)))
     , *^#(0(x), y) -> c_20(0^#(*(x, y)))
     , *^#(#(), x) -> c_21()
     , *^#(+(x, y), z) -> c_22(+^#(*(x, z), *(y, z)))
     , *^#(1(x), y) -> c_23(+^#(0(*(x, y)), y))
     , *^#(j(x), y) -> c_24(-^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_25(*^#(x, *(y, z))) }
   Strict Trs:
     { 0(#()) -> #()
     , +(x, #()) -> x
     , +(0(x), 0(y)) -> 0(+(x, y))
     , +(0(x), 1(y)) -> 1(+(x, y))
     , +(0(x), j(y)) -> j(+(x, y))
     , +(#(), x) -> x
     , +(+(x, y), z) -> +(x, +(y, z))
     , +(1(x), 0(y)) -> 1(+(x, y))
     , +(1(x), 1(y)) -> j(+(+(x, y), 1(#())))
     , +(1(x), j(y)) -> 0(+(x, y))
     , +(j(x), 0(y)) -> j(+(x, y))
     , +(j(x), 1(y)) -> 0(+(x, y))
     , +(j(x), j(y)) -> 1(+(+(x, y), j(#())))
     , opp(0(x)) -> 0(opp(x))
     , opp(#()) -> #()
     , opp(1(x)) -> j(opp(x))
     , opp(j(x)) -> 1(opp(x))
     , -(x, y) -> +(x, opp(y))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(0(x), y) -> 0(*(x, y))
     , *(#(), x) -> #()
     , *(+(x, y), z) -> +(*(x, z), *(y, z))
     , *(1(x), y) -> +(0(*(x, y)), y)
     , *(j(x), y) -> -(0(*(x, y)), y)
     , *(*(x, y), z) -> *(x, *(y, z)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,15,21} by applications
   of Pre({1,15,21}) = {2,3,6,10,12,14,16,17,20,25}. 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: +^#(0(x), j(y)) -> c_5(+^#(x, y))
       , 6: +^#(#(), x) -> c_6(x)
       , 7: +^#(+(x, y), z) -> c_7(+^#(x, +(y, z)))
       , 8: +^#(1(x), 0(y)) -> c_8(+^#(x, y))
       , 9: +^#(1(x), 1(y)) -> c_9(+^#(+(x, y), 1(#())))
       , 10: +^#(1(x), j(y)) -> c_10(0^#(+(x, y)))
       , 11: +^#(j(x), 0(y)) -> c_11(+^#(x, y))
       , 12: +^#(j(x), 1(y)) -> c_12(0^#(+(x, y)))
       , 13: +^#(j(x), j(y)) -> c_13(+^#(+(x, y), j(#())))
       , 14: opp^#(0(x)) -> c_14(0^#(opp(x)))
       , 15: opp^#(#()) -> c_15()
       , 16: opp^#(1(x)) -> c_16(opp^#(x))
       , 17: opp^#(j(x)) -> c_17(opp^#(x))
       , 18: -^#(x, y) -> c_18(+^#(x, opp(y)))
       , 19: *^#(x, +(y, z)) -> c_19(+^#(*(x, y), *(x, z)))
       , 20: *^#(0(x), y) -> c_20(0^#(*(x, y)))
       , 21: *^#(#(), x) -> c_21()
       , 22: *^#(+(x, y), z) -> c_22(+^#(*(x, z), *(y, z)))
       , 23: *^#(1(x), y) -> c_23(+^#(0(*(x, y)), y))
       , 24: *^#(j(x), y) -> c_24(-^#(0(*(x, y)), y))
       , 25: *^#(*(x, y), z) -> c_25(*^#(x, *(y, z))) }
   
   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))
     , +^#(0(x), j(y)) -> c_5(+^#(x, y))
     , +^#(#(), x) -> c_6(x)
     , +^#(+(x, y), z) -> c_7(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_8(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_9(+^#(+(x, y), 1(#())))
     , +^#(1(x), j(y)) -> c_10(0^#(+(x, y)))
     , +^#(j(x), 0(y)) -> c_11(+^#(x, y))
     , +^#(j(x), 1(y)) -> c_12(0^#(+(x, y)))
     , +^#(j(x), j(y)) -> c_13(+^#(+(x, y), j(#())))
     , opp^#(0(x)) -> c_14(0^#(opp(x)))
     , opp^#(1(x)) -> c_16(opp^#(x))
     , opp^#(j(x)) -> c_17(opp^#(x))
     , -^#(x, y) -> c_18(+^#(x, opp(y)))
     , *^#(x, +(y, z)) -> c_19(+^#(*(x, y), *(x, z)))
     , *^#(0(x), y) -> c_20(0^#(*(x, y)))
     , *^#(+(x, y), z) -> c_22(+^#(*(x, z), *(y, z)))
     , *^#(1(x), y) -> c_23(+^#(0(*(x, y)), y))
     , *^#(j(x), y) -> c_24(-^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_25(*^#(x, *(y, z))) }
   Strict Trs:
     { 0(#()) -> #()
     , +(x, #()) -> x
     , +(0(x), 0(y)) -> 0(+(x, y))
     , +(0(x), 1(y)) -> 1(+(x, y))
     , +(0(x), j(y)) -> j(+(x, y))
     , +(#(), x) -> x
     , +(+(x, y), z) -> +(x, +(y, z))
     , +(1(x), 0(y)) -> 1(+(x, y))
     , +(1(x), 1(y)) -> j(+(+(x, y), 1(#())))
     , +(1(x), j(y)) -> 0(+(x, y))
     , +(j(x), 0(y)) -> j(+(x, y))
     , +(j(x), 1(y)) -> 0(+(x, y))
     , +(j(x), j(y)) -> 1(+(+(x, y), j(#())))
     , opp(0(x)) -> 0(opp(x))
     , opp(#()) -> #()
     , opp(1(x)) -> j(opp(x))
     , opp(j(x)) -> 1(opp(x))
     , -(x, y) -> +(x, opp(y))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(0(x), y) -> 0(*(x, y))
     , *(#(), x) -> #()
     , *(+(x, y), z) -> +(*(x, z), *(y, z))
     , *(1(x), y) -> +(0(*(x, y)), y)
     , *(j(x), y) -> -(0(*(x, y)), y)
     , *(*(x, y), z) -> *(x, *(y, z)) }
   Weak DPs:
     { 0^#(#()) -> c_1()
     , opp^#(#()) -> c_15()
     , *^#(#(), x) -> c_21() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,9,11,13,18} by
   applications of Pre({2,9,11,13,18}) =
   {1,3,4,5,6,7,8,10,12,14,15,16,17,19,20,22}. 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: +^#(0(x), j(y)) -> c_5(+^#(x, y))
       , 5: +^#(#(), x) -> c_6(x)
       , 6: +^#(+(x, y), z) -> c_7(+^#(x, +(y, z)))
       , 7: +^#(1(x), 0(y)) -> c_8(+^#(x, y))
       , 8: +^#(1(x), 1(y)) -> c_9(+^#(+(x, y), 1(#())))
       , 9: +^#(1(x), j(y)) -> c_10(0^#(+(x, y)))
       , 10: +^#(j(x), 0(y)) -> c_11(+^#(x, y))
       , 11: +^#(j(x), 1(y)) -> c_12(0^#(+(x, y)))
       , 12: +^#(j(x), j(y)) -> c_13(+^#(+(x, y), j(#())))
       , 13: opp^#(0(x)) -> c_14(0^#(opp(x)))
       , 14: opp^#(1(x)) -> c_16(opp^#(x))
       , 15: opp^#(j(x)) -> c_17(opp^#(x))
       , 16: -^#(x, y) -> c_18(+^#(x, opp(y)))
       , 17: *^#(x, +(y, z)) -> c_19(+^#(*(x, y), *(x, z)))
       , 18: *^#(0(x), y) -> c_20(0^#(*(x, y)))
       , 19: *^#(+(x, y), z) -> c_22(+^#(*(x, z), *(y, z)))
       , 20: *^#(1(x), y) -> c_23(+^#(0(*(x, y)), y))
       , 21: *^#(j(x), y) -> c_24(-^#(0(*(x, y)), y))
       , 22: *^#(*(x, y), z) -> c_25(*^#(x, *(y, z)))
       , 23: 0^#(#()) -> c_1()
       , 24: opp^#(#()) -> c_15()
       , 25: *^#(#(), x) -> c_21() }
   
   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))
     , +^#(0(x), j(y)) -> c_5(+^#(x, y))
     , +^#(#(), x) -> c_6(x)
     , +^#(+(x, y), z) -> c_7(+^#(x, +(y, z)))
     , +^#(1(x), 0(y)) -> c_8(+^#(x, y))
     , +^#(1(x), 1(y)) -> c_9(+^#(+(x, y), 1(#())))
     , +^#(j(x), 0(y)) -> c_11(+^#(x, y))
     , +^#(j(x), j(y)) -> c_13(+^#(+(x, y), j(#())))
     , opp^#(1(x)) -> c_16(opp^#(x))
     , opp^#(j(x)) -> c_17(opp^#(x))
     , -^#(x, y) -> c_18(+^#(x, opp(y)))
     , *^#(x, +(y, z)) -> c_19(+^#(*(x, y), *(x, z)))
     , *^#(+(x, y), z) -> c_22(+^#(*(x, z), *(y, z)))
     , *^#(1(x), y) -> c_23(+^#(0(*(x, y)), y))
     , *^#(j(x), y) -> c_24(-^#(0(*(x, y)), y))
     , *^#(*(x, y), z) -> c_25(*^#(x, *(y, z))) }
   Strict Trs:
     { 0(#()) -> #()
     , +(x, #()) -> x
     , +(0(x), 0(y)) -> 0(+(x, y))
     , +(0(x), 1(y)) -> 1(+(x, y))
     , +(0(x), j(y)) -> j(+(x, y))
     , +(#(), x) -> x
     , +(+(x, y), z) -> +(x, +(y, z))
     , +(1(x), 0(y)) -> 1(+(x, y))
     , +(1(x), 1(y)) -> j(+(+(x, y), 1(#())))
     , +(1(x), j(y)) -> 0(+(x, y))
     , +(j(x), 0(y)) -> j(+(x, y))
     , +(j(x), 1(y)) -> 0(+(x, y))
     , +(j(x), j(y)) -> 1(+(+(x, y), j(#())))
     , opp(0(x)) -> 0(opp(x))
     , opp(#()) -> #()
     , opp(1(x)) -> j(opp(x))
     , opp(j(x)) -> 1(opp(x))
     , -(x, y) -> +(x, opp(y))
     , *(x, +(y, z)) -> +(*(x, y), *(x, z))
     , *(0(x), y) -> 0(*(x, y))
     , *(#(), x) -> #()
     , *(+(x, y), z) -> +(*(x, z), *(y, z))
     , *(1(x), y) -> +(0(*(x, y)), y)
     , *(j(x), y) -> -(0(*(x, y)), y)
     , *(*(x, y), z) -> *(x, *(y, z)) }
   Weak DPs:
     { 0^#(#()) -> c_1()
     , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y)))
     , +^#(1(x), j(y)) -> c_10(0^#(+(x, y)))
     , +^#(j(x), 1(y)) -> c_12(0^#(+(x, y)))
     , opp^#(0(x)) -> c_14(0^#(opp(x)))
     , opp^#(#()) -> c_15()
     , *^#(0(x), y) -> c_20(0^#(*(x, y)))
     , *^#(#(), x) -> c_21() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..