MAYBE

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

Strict Trs:
  { div(x, y) -> div2(x, y, 0())
  , div2(x, y, i) ->
    if1(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i))
  , if1(true(), b, x, y, i, j) -> divZeroError()
  , if1(false(), b, x, y, i, j) -> if2(b, x, y, i, j)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , plus(x, y) -> plusIter(x, y, 0())
  , inc(0()) -> 0()
  , inc(s(i)) -> s(inc(i))
  , if2(true(), x, y, i, j) -> div2(minus(x, y), y, j)
  , if2(false(), x, y, i, j) -> i
  , minus(x, 0()) -> x
  , minus(0(), y) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
  , ifPlus(true(), x, y, z) -> y
  , ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
  , a() -> c()
  , a() -> d() }
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:
     { div^#(x, y) -> c_1(div2^#(x, y, 0()))
     , div2^#(x, y, i) ->
       c_2(if1^#(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i)))
     , if1^#(true(), b, x, y, i, j) -> c_3()
     , if1^#(false(), b, x, y, i, j) -> c_4(if2^#(b, x, y, i, j))
     , if2^#(true(), x, y, i, j) -> c_11(div2^#(minus(x, y), y, j))
     , if2^#(false(), x, y, i, j) -> c_12(i)
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , plus^#(x, y) -> c_8(plusIter^#(x, y, 0()))
     , plusIter^#(x, y, z) -> c_16(ifPlus^#(le(x, z), x, y, z))
     , inc^#(0()) -> c_9()
     , inc^#(s(i)) -> c_10(inc^#(i))
     , minus^#(x, 0()) -> c_13(x)
     , minus^#(0(), y) -> c_14()
     , minus^#(s(x), s(y)) -> c_15(minus^#(x, y))
     , ifPlus^#(true(), x, y, z) -> c_17(y)
     , ifPlus^#(false(), x, y, z) -> c_18(plusIter^#(x, s(y), s(z)))
     , a^#() -> c_19()
     , a^#() -> c_20() }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { div^#(x, y) -> c_1(div2^#(x, y, 0()))
     , div2^#(x, y, i) ->
       c_2(if1^#(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i)))
     , if1^#(true(), b, x, y, i, j) -> c_3()
     , if1^#(false(), b, x, y, i, j) -> c_4(if2^#(b, x, y, i, j))
     , if2^#(true(), x, y, i, j) -> c_11(div2^#(minus(x, y), y, j))
     , if2^#(false(), x, y, i, j) -> c_12(i)
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , plus^#(x, y) -> c_8(plusIter^#(x, y, 0()))
     , plusIter^#(x, y, z) -> c_16(ifPlus^#(le(x, z), x, y, z))
     , inc^#(0()) -> c_9()
     , inc^#(s(i)) -> c_10(inc^#(i))
     , minus^#(x, 0()) -> c_13(x)
     , minus^#(0(), y) -> c_14()
     , minus^#(s(x), s(y)) -> c_15(minus^#(x, y))
     , ifPlus^#(true(), x, y, z) -> c_17(y)
     , ifPlus^#(false(), x, y, z) -> c_18(plusIter^#(x, s(y), s(z)))
     , a^#() -> c_19()
     , a^#() -> c_20() }
   Strict Trs:
     { div(x, y) -> div2(x, y, 0())
     , div2(x, y, i) ->
       if1(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i))
     , if1(true(), b, x, y, i, j) -> divZeroError()
     , if1(false(), b, x, y, i, j) -> if2(b, x, y, i, j)
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , plus(x, y) -> plusIter(x, y, 0())
     , inc(0()) -> 0()
     , inc(s(i)) -> s(inc(i))
     , if2(true(), x, y, i, j) -> div2(minus(x, y), y, j)
     , if2(false(), x, y, i, j) -> i
     , minus(x, 0()) -> x
     , minus(0(), y) -> 0()
     , minus(s(x), s(y)) -> minus(x, y)
     , plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
     , ifPlus(true(), x, y, z) -> y
     , ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
     , a() -> c()
     , a() -> d() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {3,7,8,12,15,19,20} by
   applications of Pre({3,7,8,12,15,19,20}) = {2,6,9,13,14,16,17}.
   Here rules are labeled as follows:
   
     DPs:
       { 1: div^#(x, y) -> c_1(div2^#(x, y, 0()))
       , 2: div2^#(x, y, i) ->
            c_2(if1^#(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i)))
       , 3: if1^#(true(), b, x, y, i, j) -> c_3()
       , 4: if1^#(false(), b, x, y, i, j) -> c_4(if2^#(b, x, y, i, j))
       , 5: if2^#(true(), x, y, i, j) -> c_11(div2^#(minus(x, y), y, j))
       , 6: if2^#(false(), x, y, i, j) -> c_12(i)
       , 7: le^#(0(), y) -> c_5()
       , 8: le^#(s(x), 0()) -> c_6()
       , 9: le^#(s(x), s(y)) -> c_7(le^#(x, y))
       , 10: plus^#(x, y) -> c_8(plusIter^#(x, y, 0()))
       , 11: plusIter^#(x, y, z) -> c_16(ifPlus^#(le(x, z), x, y, z))
       , 12: inc^#(0()) -> c_9()
       , 13: inc^#(s(i)) -> c_10(inc^#(i))
       , 14: minus^#(x, 0()) -> c_13(x)
       , 15: minus^#(0(), y) -> c_14()
       , 16: minus^#(s(x), s(y)) -> c_15(minus^#(x, y))
       , 17: ifPlus^#(true(), x, y, z) -> c_17(y)
       , 18: ifPlus^#(false(), x, y, z) -> c_18(plusIter^#(x, s(y), s(z)))
       , 19: a^#() -> c_19()
       , 20: a^#() -> c_20() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { div^#(x, y) -> c_1(div2^#(x, y, 0()))
     , div2^#(x, y, i) ->
       c_2(if1^#(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i)))
     , if1^#(false(), b, x, y, i, j) -> c_4(if2^#(b, x, y, i, j))
     , if2^#(true(), x, y, i, j) -> c_11(div2^#(minus(x, y), y, j))
     , if2^#(false(), x, y, i, j) -> c_12(i)
     , le^#(s(x), s(y)) -> c_7(le^#(x, y))
     , plus^#(x, y) -> c_8(plusIter^#(x, y, 0()))
     , plusIter^#(x, y, z) -> c_16(ifPlus^#(le(x, z), x, y, z))
     , inc^#(s(i)) -> c_10(inc^#(i))
     , minus^#(x, 0()) -> c_13(x)
     , minus^#(s(x), s(y)) -> c_15(minus^#(x, y))
     , ifPlus^#(true(), x, y, z) -> c_17(y)
     , ifPlus^#(false(), x, y, z) -> c_18(plusIter^#(x, s(y), s(z))) }
   Strict Trs:
     { div(x, y) -> div2(x, y, 0())
     , div2(x, y, i) ->
       if1(le(y, 0()), le(y, x), x, y, plus(i, 0()), inc(i))
     , if1(true(), b, x, y, i, j) -> divZeroError()
     , if1(false(), b, x, y, i, j) -> if2(b, x, y, i, j)
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , plus(x, y) -> plusIter(x, y, 0())
     , inc(0()) -> 0()
     , inc(s(i)) -> s(inc(i))
     , if2(true(), x, y, i, j) -> div2(minus(x, y), y, j)
     , if2(false(), x, y, i, j) -> i
     , minus(x, 0()) -> x
     , minus(0(), y) -> 0()
     , minus(s(x), s(y)) -> minus(x, y)
     , plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
     , ifPlus(true(), x, y, z) -> y
     , ifPlus(false(), x, y, z) -> plusIter(x, s(y), s(z))
     , a() -> c()
     , a() -> d() }
   Weak DPs:
     { if1^#(true(), b, x, y, i, j) -> c_3()
     , le^#(0(), y) -> c_5()
     , le^#(s(x), 0()) -> c_6()
     , inc^#(0()) -> c_9()
     , minus^#(0(), y) -> c_14()
     , a^#() -> c_19()
     , a^#() -> c_20() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..