MAYBE

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

Strict Trs:
  { inc(s(x)) -> s(inc(x))
  , inc(0()) -> s(0())
  , plus(x, y) -> ifPlus(eq(x, 0()), minus(x, s(0())), x, inc(x))
  , ifPlus(false(), x, y, z) -> plus(x, z)
  , ifPlus(true(), x, y, z) -> y
  , eq(x, x) -> true()
  , eq(s(x), s(y)) -> eq(x, y)
  , eq(s(x), 0()) -> false()
  , eq(0(), s(x)) -> false()
  , eq(0(), 0()) -> true()
  , minus(x, x) -> 0()
  , minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , minus(0(), x) -> 0()
  , times(x, y) -> timesIter(x, y, 0())
  , timesIter(x, y, z) ->
    ifTimes(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z))
  , ifTimes(false(), x, y, z, u) -> timesIter(x, y, u)
  , ifTimes(true(), x, y, z, u) -> z
  , f() -> g()
  , f() -> h() }
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:
     { inc^#(s(x)) -> c_1(inc^#(x))
     , inc^#(0()) -> c_2()
     , plus^#(x, y) ->
       c_3(ifPlus^#(eq(x, 0()), minus(x, s(0())), x, inc(x)))
     , ifPlus^#(false(), x, y, z) -> c_4(plus^#(x, z))
     , ifPlus^#(true(), x, y, z) -> c_5(y)
     , eq^#(x, x) -> c_6()
     , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
     , eq^#(s(x), 0()) -> c_8()
     , eq^#(0(), s(x)) -> c_9()
     , eq^#(0(), 0()) -> c_10()
     , minus^#(x, x) -> c_11()
     , minus^#(x, 0()) -> c_12(x)
     , minus^#(s(x), s(y)) -> c_13(minus^#(x, y))
     , minus^#(0(), x) -> c_14()
     , times^#(x, y) -> c_15(timesIter^#(x, y, 0()))
     , timesIter^#(x, y, z) ->
       c_16(ifTimes^#(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z)))
     , ifTimes^#(false(), x, y, z, u) -> c_17(timesIter^#(x, y, u))
     , ifTimes^#(true(), x, y, z, u) -> c_18(z)
     , f^#() -> c_19()
     , f^#() -> c_20() }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { inc^#(s(x)) -> c_1(inc^#(x))
     , inc^#(0()) -> c_2()
     , plus^#(x, y) ->
       c_3(ifPlus^#(eq(x, 0()), minus(x, s(0())), x, inc(x)))
     , ifPlus^#(false(), x, y, z) -> c_4(plus^#(x, z))
     , ifPlus^#(true(), x, y, z) -> c_5(y)
     , eq^#(x, x) -> c_6()
     , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
     , eq^#(s(x), 0()) -> c_8()
     , eq^#(0(), s(x)) -> c_9()
     , eq^#(0(), 0()) -> c_10()
     , minus^#(x, x) -> c_11()
     , minus^#(x, 0()) -> c_12(x)
     , minus^#(s(x), s(y)) -> c_13(minus^#(x, y))
     , minus^#(0(), x) -> c_14()
     , times^#(x, y) -> c_15(timesIter^#(x, y, 0()))
     , timesIter^#(x, y, z) ->
       c_16(ifTimes^#(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z)))
     , ifTimes^#(false(), x, y, z, u) -> c_17(timesIter^#(x, y, u))
     , ifTimes^#(true(), x, y, z, u) -> c_18(z)
     , f^#() -> c_19()
     , f^#() -> c_20() }
   Strict Trs:
     { inc(s(x)) -> s(inc(x))
     , inc(0()) -> s(0())
     , plus(x, y) -> ifPlus(eq(x, 0()), minus(x, s(0())), x, inc(x))
     , ifPlus(false(), x, y, z) -> plus(x, z)
     , ifPlus(true(), x, y, z) -> y
     , eq(x, x) -> true()
     , eq(s(x), s(y)) -> eq(x, y)
     , eq(s(x), 0()) -> false()
     , eq(0(), s(x)) -> false()
     , eq(0(), 0()) -> true()
     , minus(x, x) -> 0()
     , minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , minus(0(), x) -> 0()
     , times(x, y) -> timesIter(x, y, 0())
     , timesIter(x, y, z) ->
       ifTimes(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z))
     , ifTimes(false(), x, y, z, u) -> timesIter(x, y, u)
     , ifTimes(true(), x, y, z, u) -> z
     , f() -> g()
     , f() -> h() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,6,8,9,10,11,14,19,20}
   by applications of Pre({2,6,8,9,10,11,14,19,20}) =
   {1,5,7,12,13,18}. Here rules are labeled as follows:
   
     DPs:
       { 1: inc^#(s(x)) -> c_1(inc^#(x))
       , 2: inc^#(0()) -> c_2()
       , 3: plus^#(x, y) ->
            c_3(ifPlus^#(eq(x, 0()), minus(x, s(0())), x, inc(x)))
       , 4: ifPlus^#(false(), x, y, z) -> c_4(plus^#(x, z))
       , 5: ifPlus^#(true(), x, y, z) -> c_5(y)
       , 6: eq^#(x, x) -> c_6()
       , 7: eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
       , 8: eq^#(s(x), 0()) -> c_8()
       , 9: eq^#(0(), s(x)) -> c_9()
       , 10: eq^#(0(), 0()) -> c_10()
       , 11: minus^#(x, x) -> c_11()
       , 12: minus^#(x, 0()) -> c_12(x)
       , 13: minus^#(s(x), s(y)) -> c_13(minus^#(x, y))
       , 14: minus^#(0(), x) -> c_14()
       , 15: times^#(x, y) -> c_15(timesIter^#(x, y, 0()))
       , 16: timesIter^#(x, y, z) ->
             c_16(ifTimes^#(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z)))
       , 17: ifTimes^#(false(), x, y, z, u) -> c_17(timesIter^#(x, y, u))
       , 18: ifTimes^#(true(), x, y, z, u) -> c_18(z)
       , 19: f^#() -> c_19()
       , 20: f^#() -> c_20() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { inc^#(s(x)) -> c_1(inc^#(x))
     , plus^#(x, y) ->
       c_3(ifPlus^#(eq(x, 0()), minus(x, s(0())), x, inc(x)))
     , ifPlus^#(false(), x, y, z) -> c_4(plus^#(x, z))
     , ifPlus^#(true(), x, y, z) -> c_5(y)
     , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
     , minus^#(x, 0()) -> c_12(x)
     , minus^#(s(x), s(y)) -> c_13(minus^#(x, y))
     , times^#(x, y) -> c_15(timesIter^#(x, y, 0()))
     , timesIter^#(x, y, z) ->
       c_16(ifTimes^#(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z)))
     , ifTimes^#(false(), x, y, z, u) -> c_17(timesIter^#(x, y, u))
     , ifTimes^#(true(), x, y, z, u) -> c_18(z) }
   Strict Trs:
     { inc(s(x)) -> s(inc(x))
     , inc(0()) -> s(0())
     , plus(x, y) -> ifPlus(eq(x, 0()), minus(x, s(0())), x, inc(x))
     , ifPlus(false(), x, y, z) -> plus(x, z)
     , ifPlus(true(), x, y, z) -> y
     , eq(x, x) -> true()
     , eq(s(x), s(y)) -> eq(x, y)
     , eq(s(x), 0()) -> false()
     , eq(0(), s(x)) -> false()
     , eq(0(), 0()) -> true()
     , minus(x, x) -> 0()
     , minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , minus(0(), x) -> 0()
     , times(x, y) -> timesIter(x, y, 0())
     , timesIter(x, y, z) ->
       ifTimes(eq(x, 0()), minus(x, s(0())), y, z, plus(y, z))
     , ifTimes(false(), x, y, z, u) -> timesIter(x, y, u)
     , ifTimes(true(), x, y, z, u) -> z
     , f() -> g()
     , f() -> h() }
   Weak DPs:
     { inc^#(0()) -> c_2()
     , eq^#(x, x) -> c_6()
     , eq^#(s(x), 0()) -> c_8()
     , eq^#(0(), s(x)) -> c_9()
     , eq^#(0(), 0()) -> c_10()
     , minus^#(x, x) -> c_11()
     , minus^#(0(), x) -> c_14()
     , f^#() -> c_19()
     , f^#() -> c_20() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..