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, #()) -> x
  , -(0(x), 0(y)) -> 0(-(x, y))
  , -(0(x), 1(y)) -> 1(-(-(x, y), 1(#())))
  , -(#(), x) -> #()
  , -(1(x), 0(y)) -> 1(-(x, y))
  , -(1(x), 1(y)) -> 0(-(x, y))
  , not(true()) -> false()
  , not(false()) -> true()
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , ge(x, #()) -> true()
  , ge(0(x), 0(y)) -> ge(x, y)
  , ge(0(x), 1(y)) -> not(ge(y, x))
  , ge(#(), 0(x)) -> ge(#(), x)
  , ge(#(), 1(x)) -> false()
  , ge(1(x), 0(y)) -> ge(x, y)
  , ge(1(x), 1(y)) -> ge(x, y)
  , log(x) -> -(log'(x), 1(#()))
  , log'(0(x)) -> if(ge(x, 1(#())), +(log'(x), 1(#())), #())
  , log'(#()) -> #()
  , log'(1(x)) -> +(log'(x), 1(#())) }
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) '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) '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, #()) -> c_9(x)
     , -^#(0(x), 0(y)) -> c_10(0^#(-(x, y)))
     , -^#(0(x), 1(y)) -> c_11(-^#(-(x, y), 1(#())))
     , -^#(#(), x) -> c_12()
     , -^#(1(x), 0(y)) -> c_13(-^#(x, y))
     , -^#(1(x), 1(y)) -> c_14(0^#(-(x, y)))
     , not^#(true()) -> c_15()
     , not^#(false()) -> c_16()
     , if^#(true(), x, y) -> c_17(x)
     , if^#(false(), x, y) -> c_18(y)
     , ge^#(x, #()) -> c_19()
     , ge^#(0(x), 0(y)) -> c_20(ge^#(x, y))
     , ge^#(0(x), 1(y)) -> c_21(not^#(ge(y, x)))
     , ge^#(#(), 0(x)) -> c_22(ge^#(#(), x))
     , ge^#(#(), 1(x)) -> c_23()
     , ge^#(1(x), 0(y)) -> c_24(ge^#(x, y))
     , ge^#(1(x), 1(y)) -> c_25(ge^#(x, y))
     , log^#(x) -> c_26(-^#(log'(x), 1(#())))
     , log'^#(0(x)) ->
       c_27(if^#(ge(x, 1(#())), +(log'(x), 1(#())), #()))
     , log'^#(#()) -> c_28()
     , log'^#(1(x)) -> c_29(+^#(log'(x), 1(#()))) }
   
   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, #()) -> c_9(x)
     , -^#(0(x), 0(y)) -> c_10(0^#(-(x, y)))
     , -^#(0(x), 1(y)) -> c_11(-^#(-(x, y), 1(#())))
     , -^#(#(), x) -> c_12()
     , -^#(1(x), 0(y)) -> c_13(-^#(x, y))
     , -^#(1(x), 1(y)) -> c_14(0^#(-(x, y)))
     , not^#(true()) -> c_15()
     , not^#(false()) -> c_16()
     , if^#(true(), x, y) -> c_17(x)
     , if^#(false(), x, y) -> c_18(y)
     , ge^#(x, #()) -> c_19()
     , ge^#(0(x), 0(y)) -> c_20(ge^#(x, y))
     , ge^#(0(x), 1(y)) -> c_21(not^#(ge(y, x)))
     , ge^#(#(), 0(x)) -> c_22(ge^#(#(), x))
     , ge^#(#(), 1(x)) -> c_23()
     , ge^#(1(x), 0(y)) -> c_24(ge^#(x, y))
     , ge^#(1(x), 1(y)) -> c_25(ge^#(x, y))
     , log^#(x) -> c_26(-^#(log'(x), 1(#())))
     , log'^#(0(x)) ->
       c_27(if^#(ge(x, 1(#())), +(log'(x), 1(#())), #()))
     , log'^#(#()) -> c_28()
     , log'^#(1(x)) -> c_29(+^#(log'(x), 1(#()))) }
   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, #()) -> x
     , -(0(x), 0(y)) -> 0(-(x, y))
     , -(0(x), 1(y)) -> 1(-(-(x, y), 1(#())))
     , -(#(), x) -> #()
     , -(1(x), 0(y)) -> 1(-(x, y))
     , -(1(x), 1(y)) -> 0(-(x, y))
     , not(true()) -> false()
     , not(false()) -> true()
     , if(true(), x, y) -> x
     , if(false(), x, y) -> y
     , ge(x, #()) -> true()
     , ge(0(x), 0(y)) -> ge(x, y)
     , ge(0(x), 1(y)) -> not(ge(y, x))
     , ge(#(), 0(x)) -> ge(#(), x)
     , ge(#(), 1(x)) -> false()
     , ge(1(x), 0(y)) -> ge(x, y)
     , ge(1(x), 1(y)) -> ge(x, y)
     , log(x) -> -(log'(x), 1(#()))
     , log'(0(x)) -> if(ge(x, 1(#())), +(log'(x), 1(#())), #())
     , log'(#()) -> #()
     , log'(1(x)) -> +(log'(x), 1(#())) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,12,15,16,19,23,28} by
   applications of Pre({1,12,15,16,19,23,28}) =
   {2,3,5,8,9,10,11,13,14,17,18,20,21,22,24,25,26}. 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, #()) -> c_9(x)
       , 10: -^#(0(x), 0(y)) -> c_10(0^#(-(x, y)))
       , 11: -^#(0(x), 1(y)) -> c_11(-^#(-(x, y), 1(#())))
       , 12: -^#(#(), x) -> c_12()
       , 13: -^#(1(x), 0(y)) -> c_13(-^#(x, y))
       , 14: -^#(1(x), 1(y)) -> c_14(0^#(-(x, y)))
       , 15: not^#(true()) -> c_15()
       , 16: not^#(false()) -> c_16()
       , 17: if^#(true(), x, y) -> c_17(x)
       , 18: if^#(false(), x, y) -> c_18(y)
       , 19: ge^#(x, #()) -> c_19()
       , 20: ge^#(0(x), 0(y)) -> c_20(ge^#(x, y))
       , 21: ge^#(0(x), 1(y)) -> c_21(not^#(ge(y, x)))
       , 22: ge^#(#(), 0(x)) -> c_22(ge^#(#(), x))
       , 23: ge^#(#(), 1(x)) -> c_23()
       , 24: ge^#(1(x), 0(y)) -> c_24(ge^#(x, y))
       , 25: ge^#(1(x), 1(y)) -> c_25(ge^#(x, y))
       , 26: log^#(x) -> c_26(-^#(log'(x), 1(#())))
       , 27: log'^#(0(x)) ->
             c_27(if^#(ge(x, 1(#())), +(log'(x), 1(#())), #()))
       , 28: log'^#(#()) -> c_28()
       , 29: log'^#(1(x)) -> c_29(+^#(log'(x), 1(#()))) }
   
   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, #()) -> c_9(x)
     , -^#(0(x), 0(y)) -> c_10(0^#(-(x, y)))
     , -^#(0(x), 1(y)) -> c_11(-^#(-(x, y), 1(#())))
     , -^#(1(x), 0(y)) -> c_13(-^#(x, y))
     , -^#(1(x), 1(y)) -> c_14(0^#(-(x, y)))
     , if^#(true(), x, y) -> c_17(x)
     , if^#(false(), x, y) -> c_18(y)
     , ge^#(0(x), 0(y)) -> c_20(ge^#(x, y))
     , ge^#(0(x), 1(y)) -> c_21(not^#(ge(y, x)))
     , ge^#(#(), 0(x)) -> c_22(ge^#(#(), x))
     , ge^#(1(x), 0(y)) -> c_24(ge^#(x, y))
     , ge^#(1(x), 1(y)) -> c_25(ge^#(x, y))
     , log^#(x) -> c_26(-^#(log'(x), 1(#())))
     , log'^#(0(x)) ->
       c_27(if^#(ge(x, 1(#())), +(log'(x), 1(#())), #()))
     , log'^#(1(x)) -> c_29(+^#(log'(x), 1(#()))) }
   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, #()) -> x
     , -(0(x), 0(y)) -> 0(-(x, y))
     , -(0(x), 1(y)) -> 1(-(-(x, y), 1(#())))
     , -(#(), x) -> #()
     , -(1(x), 0(y)) -> 1(-(x, y))
     , -(1(x), 1(y)) -> 0(-(x, y))
     , not(true()) -> false()
     , not(false()) -> true()
     , if(true(), x, y) -> x
     , if(false(), x, y) -> y
     , ge(x, #()) -> true()
     , ge(0(x), 0(y)) -> ge(x, y)
     , ge(0(x), 1(y)) -> not(ge(y, x))
     , ge(#(), 0(x)) -> ge(#(), x)
     , ge(#(), 1(x)) -> false()
     , ge(1(x), 0(y)) -> ge(x, y)
     , ge(1(x), 1(y)) -> ge(x, y)
     , log(x) -> -(log'(x), 1(#()))
     , log'(0(x)) -> if(ge(x, 1(#())), +(log'(x), 1(#())), #())
     , log'(#()) -> #()
     , log'(1(x)) -> +(log'(x), 1(#())) }
   Weak DPs:
     { 0^#(#()) -> c_1()
     , -^#(#(), x) -> c_12()
     , not^#(true()) -> c_15()
     , not^#(false()) -> c_16()
     , ge^#(x, #()) -> c_19()
     , ge^#(#(), 1(x)) -> c_23()
     , log'^#(#()) -> c_28() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,7,9,12,16} by
   applications of Pre({2,7,9,12,16}) =
   {1,3,4,5,6,8,10,11,13,14,15,18,19,20,22}. 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: +^#(#(), 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, #()) -> c_9(x)
       , 9: -^#(0(x), 0(y)) -> c_10(0^#(-(x, y)))
       , 10: -^#(0(x), 1(y)) -> c_11(-^#(-(x, y), 1(#())))
       , 11: -^#(1(x), 0(y)) -> c_13(-^#(x, y))
       , 12: -^#(1(x), 1(y)) -> c_14(0^#(-(x, y)))
       , 13: if^#(true(), x, y) -> c_17(x)
       , 14: if^#(false(), x, y) -> c_18(y)
       , 15: ge^#(0(x), 0(y)) -> c_20(ge^#(x, y))
       , 16: ge^#(0(x), 1(y)) -> c_21(not^#(ge(y, x)))
       , 17: ge^#(#(), 0(x)) -> c_22(ge^#(#(), x))
       , 18: ge^#(1(x), 0(y)) -> c_24(ge^#(x, y))
       , 19: ge^#(1(x), 1(y)) -> c_25(ge^#(x, y))
       , 20: log^#(x) -> c_26(-^#(log'(x), 1(#())))
       , 21: log'^#(0(x)) ->
             c_27(if^#(ge(x, 1(#())), +(log'(x), 1(#())), #()))
       , 22: log'^#(1(x)) -> c_29(+^#(log'(x), 1(#())))
       , 23: 0^#(#()) -> c_1()
       , 24: -^#(#(), x) -> c_12()
       , 25: not^#(true()) -> c_15()
       , 26: not^#(false()) -> c_16()
       , 27: ge^#(x, #()) -> c_19()
       , 28: ge^#(#(), 1(x)) -> c_23()
       , 29: log'^#(#()) -> c_28() }
   
   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, #()) -> c_9(x)
     , -^#(0(x), 1(y)) -> c_11(-^#(-(x, y), 1(#())))
     , -^#(1(x), 0(y)) -> c_13(-^#(x, y))
     , if^#(true(), x, y) -> c_17(x)
     , if^#(false(), x, y) -> c_18(y)
     , ge^#(0(x), 0(y)) -> c_20(ge^#(x, y))
     , ge^#(#(), 0(x)) -> c_22(ge^#(#(), x))
     , ge^#(1(x), 0(y)) -> c_24(ge^#(x, y))
     , ge^#(1(x), 1(y)) -> c_25(ge^#(x, y))
     , log^#(x) -> c_26(-^#(log'(x), 1(#())))
     , log'^#(0(x)) ->
       c_27(if^#(ge(x, 1(#())), +(log'(x), 1(#())), #()))
     , log'^#(1(x)) -> c_29(+^#(log'(x), 1(#()))) }
   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, #()) -> x
     , -(0(x), 0(y)) -> 0(-(x, y))
     , -(0(x), 1(y)) -> 1(-(-(x, y), 1(#())))
     , -(#(), x) -> #()
     , -(1(x), 0(y)) -> 1(-(x, y))
     , -(1(x), 1(y)) -> 0(-(x, y))
     , not(true()) -> false()
     , not(false()) -> true()
     , if(true(), x, y) -> x
     , if(false(), x, y) -> y
     , ge(x, #()) -> true()
     , ge(0(x), 0(y)) -> ge(x, y)
     , ge(0(x), 1(y)) -> not(ge(y, x))
     , ge(#(), 0(x)) -> ge(#(), x)
     , ge(#(), 1(x)) -> false()
     , ge(1(x), 0(y)) -> ge(x, y)
     , ge(1(x), 1(y)) -> ge(x, y)
     , log(x) -> -(log'(x), 1(#()))
     , log'(0(x)) -> if(ge(x, 1(#())), +(log'(x), 1(#())), #())
     , log'(#()) -> #()
     , log'(1(x)) -> +(log'(x), 1(#())) }
   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), 0(y)) -> c_10(0^#(-(x, y)))
     , -^#(#(), x) -> c_12()
     , -^#(1(x), 1(y)) -> c_14(0^#(-(x, y)))
     , not^#(true()) -> c_15()
     , not^#(false()) -> c_16()
     , ge^#(x, #()) -> c_19()
     , ge^#(0(x), 1(y)) -> c_21(not^#(ge(y, x)))
     , ge^#(#(), 1(x)) -> c_23()
     , log'^#(#()) -> c_28() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..