MAYBE

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

Strict Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x))
  , log(x) -> logIter(x, 0())
  , logIter(x, y) ->
    if(le(s(0()), x), le(s(s(0())), x), quot(x, s(s(0()))), inc(y))
  , if(true(), true(), x, y) -> logIter(x, y)
  , if(true(), false(), x, s(y)) -> y
  , if(false(), b, x, y) -> logZeroError() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(0(), s(y)) -> c_3()
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , inc^#(0()) -> c_8()
  , inc^#(s(x)) -> c_9(inc^#(x))
  , log^#(x) -> c_10(logIter^#(x, 0()))
  , logIter^#(x, y) ->
    c_11(if^#(le(s(0()), x),
              le(s(s(0())), x),
              quot(x, s(s(0()))),
              inc(y)),
         le^#(s(0()), x),
         le^#(s(s(0())), x),
         quot^#(x, s(s(0()))),
         inc^#(y))
  , if^#(true(), true(), x, y) -> c_12(logIter^#(x, y))
  , if^#(true(), false(), x, s(y)) -> c_13()
  , if^#(false(), b, x, y) -> c_14() }

and mark the set of starting terms.

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

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(0(), s(y)) -> c_3()
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , inc^#(0()) -> c_8()
  , inc^#(s(x)) -> c_9(inc^#(x))
  , log^#(x) -> c_10(logIter^#(x, 0()))
  , logIter^#(x, y) ->
    c_11(if^#(le(s(0()), x),
              le(s(s(0())), x),
              quot(x, s(s(0()))),
              inc(y)),
         le^#(s(0()), x),
         le^#(s(s(0())), x),
         quot^#(x, s(s(0()))),
         inc^#(y))
  , if^#(true(), true(), x, y) -> c_12(logIter^#(x, y))
  , if^#(true(), false(), x, s(y)) -> c_13()
  , if^#(false(), b, x, y) -> c_14() }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x))
  , log(x) -> logIter(x, 0())
  , logIter(x, y) ->
    if(le(s(0()), x), le(s(s(0())), x), quot(x, s(s(0()))), inc(y))
  , if(true(), true(), x, y) -> logIter(x, y)
  , if(true(), false(), x, s(y)) -> y
  , if(false(), b, x, y) -> logZeroError() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,6,8,13,14} by
applications of Pre({1,3,5,6,8,13,14}) = {2,4,7,9,11}. Here rules
are labeled as follows:

  DPs:
    { 1: minus^#(x, 0()) -> c_1()
    , 2: minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
    , 3: quot^#(0(), s(y)) -> c_3()
    , 4: quot^#(s(x), s(y)) ->
         c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
    , 5: le^#(0(), y) -> c_5()
    , 6: le^#(s(x), 0()) -> c_6()
    , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y))
    , 8: inc^#(0()) -> c_8()
    , 9: inc^#(s(x)) -> c_9(inc^#(x))
    , 10: log^#(x) -> c_10(logIter^#(x, 0()))
    , 11: logIter^#(x, y) ->
          c_11(if^#(le(s(0()), x),
                    le(s(s(0())), x),
                    quot(x, s(s(0()))),
                    inc(y)),
               le^#(s(0()), x),
               le^#(s(s(0())), x),
               quot^#(x, s(s(0()))),
               inc^#(y))
    , 12: if^#(true(), true(), x, y) -> c_12(logIter^#(x, y))
    , 13: if^#(true(), false(), x, s(y)) -> c_13()
    , 14: if^#(false(), b, x, y) -> c_14() }

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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , inc^#(s(x)) -> c_9(inc^#(x))
  , log^#(x) -> c_10(logIter^#(x, 0()))
  , logIter^#(x, y) ->
    c_11(if^#(le(s(0()), x),
              le(s(s(0())), x),
              quot(x, s(s(0()))),
              inc(y)),
         le^#(s(0()), x),
         le^#(s(s(0())), x),
         quot^#(x, s(s(0()))),
         inc^#(y))
  , if^#(true(), true(), x, y) -> c_12(logIter^#(x, y)) }
Weak DPs:
  { minus^#(x, 0()) -> c_1()
  , quot^#(0(), s(y)) -> c_3()
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , inc^#(0()) -> c_8()
  , if^#(true(), false(), x, s(y)) -> c_13()
  , if^#(false(), b, x, y) -> c_14() }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x))
  , log(x) -> logIter(x, 0())
  , logIter(x, y) ->
    if(le(s(0()), x), le(s(s(0())), x), quot(x, s(s(0()))), inc(y))
  , if(true(), true(), x, y) -> logIter(x, y)
  , if(true(), false(), x, s(y)) -> y
  , if(false(), b, x, y) -> logZeroError() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ minus^#(x, 0()) -> c_1()
, quot^#(0(), s(y)) -> c_3()
, le^#(0(), y) -> c_5()
, le^#(s(x), 0()) -> c_6()
, inc^#(0()) -> c_8()
, if^#(true(), false(), x, s(y)) -> c_13()
, if^#(false(), b, x, y) -> c_14() }

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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , inc^#(s(x)) -> c_9(inc^#(x))
  , log^#(x) -> c_10(logIter^#(x, 0()))
  , logIter^#(x, y) ->
    c_11(if^#(le(s(0()), x),
              le(s(s(0())), x),
              quot(x, s(s(0()))),
              inc(y)),
         le^#(s(0()), x),
         le^#(s(s(0())), x),
         quot^#(x, s(s(0()))),
         inc^#(y))
  , if^#(true(), true(), x, y) -> c_12(logIter^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x))
  , log(x) -> logIter(x, 0())
  , logIter(x, y) ->
    if(le(s(0()), x), le(s(s(0())), x), quot(x, s(s(0()))), inc(y))
  , if(true(), true(), x, y) -> logIter(x, y)
  , if(true(), false(), x, s(y)) -> y
  , if(false(), b, x, y) -> logZeroError() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
     -->_1 minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) :1
  
  2: quot^#(s(x), s(y)) ->
     c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
     -->_1 quot^#(s(x), s(y)) ->
           c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) :2
     -->_2 minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) :1
  
  3: le^#(s(x), s(y)) -> c_7(le^#(x, y))
     -->_1 le^#(s(x), s(y)) -> c_7(le^#(x, y)) :3
  
  4: inc^#(s(x)) -> c_9(inc^#(x))
     -->_1 inc^#(s(x)) -> c_9(inc^#(x)) :4
  
  5: log^#(x) -> c_10(logIter^#(x, 0()))
     -->_1 logIter^#(x, y) ->
           c_11(if^#(le(s(0()), x),
                     le(s(s(0())), x),
                     quot(x, s(s(0()))),
                     inc(y)),
                le^#(s(0()), x),
                le^#(s(s(0())), x),
                quot^#(x, s(s(0()))),
                inc^#(y)) :6
  
  6: logIter^#(x, y) ->
     c_11(if^#(le(s(0()), x),
               le(s(s(0())), x),
               quot(x, s(s(0()))),
               inc(y)),
          le^#(s(0()), x),
          le^#(s(s(0())), x),
          quot^#(x, s(s(0()))),
          inc^#(y))
     -->_1 if^#(true(), true(), x, y) -> c_12(logIter^#(x, y)) :7
     -->_5 inc^#(s(x)) -> c_9(inc^#(x)) :4
     -->_3 le^#(s(x), s(y)) -> c_7(le^#(x, y)) :3
     -->_2 le^#(s(x), s(y)) -> c_7(le^#(x, y)) :3
     -->_4 quot^#(s(x), s(y)) ->
           c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) :2
  
  7: if^#(true(), true(), x, y) -> c_12(logIter^#(x, y))
     -->_1 logIter^#(x, y) ->
           c_11(if^#(le(s(0()), x),
                     le(s(s(0())), x),
                     quot(x, s(s(0()))),
                     inc(y)),
                le^#(s(0()), x),
                le^#(s(s(0())), x),
                quot^#(x, s(s(0()))),
                inc^#(y)) :6
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { log^#(x) -> c_10(logIter^#(x, 0())) }


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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , inc^#(s(x)) -> c_9(inc^#(x))
  , logIter^#(x, y) ->
    c_11(if^#(le(s(0()), x),
              le(s(s(0())), x),
              quot(x, s(s(0()))),
              inc(y)),
         le^#(s(0()), x),
         le^#(s(s(0())), x),
         quot^#(x, s(s(0()))),
         inc^#(y))
  , if^#(true(), true(), x, y) -> c_12(logIter^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x))
  , log(x) -> logIter(x, 0())
  , logIter(x, y) ->
    if(le(s(0()), x), le(s(s(0())), x), quot(x, s(s(0()))), inc(y))
  , if(true(), true(), x, y) -> logIter(x, y)
  , if(true(), false(), x, s(y)) -> y
  , if(false(), b, x, y) -> logZeroError() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { minus(x, 0()) -> x
    , minus(s(x), s(y)) -> minus(x, y)
    , quot(0(), s(y)) -> 0()
    , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
    , le(0(), y) -> true()
    , le(s(x), 0()) -> false()
    , le(s(x), s(y)) -> le(x, y)
    , inc(0()) -> s(0())
    , inc(s(x)) -> s(inc(x)) }

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

Strict DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , quot^#(s(x), s(y)) ->
    c_4(quot^#(minus(x, y), s(y)), minus^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , inc^#(s(x)) -> c_9(inc^#(x))
  , logIter^#(x, y) ->
    c_11(if^#(le(s(0()), x),
              le(s(s(0())), x),
              quot(x, s(s(0()))),
              inc(y)),
         le^#(s(0()), x),
         le^#(s(s(0())), x),
         quot^#(x, s(s(0()))),
         inc^#(y))
  , if^#(true(), true(), x, y) -> c_12(logIter^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , inc(0()) -> s(0())
  , inc(s(x)) -> s(inc(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..