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)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
  , if_quot(true(), x, y) -> s(quot(minus(x, y), y))
  , if_quot(false(), x, y) -> 0() }
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))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(x, s(y)) ->
    c_6(if_quot^#(le(s(y), x), x, s(y)), le^#(s(y), x))
  , if_quot^#(true(), x, y) ->
    c_7(quot^#(minus(x, y), y), minus^#(x, y))
  , if_quot^#(false(), x, y) -> c_8() }

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))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(x, s(y)) ->
    c_6(if_quot^#(le(s(y), x), x, s(y)), le^#(s(y), x))
  , if_quot^#(true(), x, y) ->
    c_7(quot^#(minus(x, y), y), minus^#(x, y))
  , if_quot^#(false(), x, y) -> c_8() }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
  , if_quot(true(), x, y) -> s(quot(minus(x, y), y))
  , if_quot(false(), x, y) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,4,8} by applications
of Pre({1,3,4,8}) = {2,5,6,7}. 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: le^#(0(), y) -> c_3()
    , 4: le^#(s(x), 0()) -> c_4()
    , 5: le^#(s(x), s(y)) -> c_5(le^#(x, y))
    , 6: quot^#(x, s(y)) ->
         c_6(if_quot^#(le(s(y), x), x, s(y)), le^#(s(y), x))
    , 7: if_quot^#(true(), x, y) ->
         c_7(quot^#(minus(x, y), y), minus^#(x, y))
    , 8: if_quot^#(false(), x, y) -> c_8() }

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))
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(x, s(y)) ->
    c_6(if_quot^#(le(s(y), x), x, s(y)), le^#(s(y), x))
  , if_quot^#(true(), x, y) ->
    c_7(quot^#(minus(x, y), y), minus^#(x, y)) }
Weak DPs:
  { minus^#(x, 0()) -> c_1()
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , if_quot^#(false(), x, y) -> c_8() }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
  , if_quot(true(), x, y) -> s(quot(minus(x, y), y))
  , if_quot(false(), x, y) -> 0() }
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()
, le^#(0(), y) -> c_3()
, le^#(s(x), 0()) -> c_4()
, if_quot^#(false(), x, y) -> c_8() }

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))
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(x, s(y)) ->
    c_6(if_quot^#(le(s(y), x), x, s(y)), le^#(s(y), x))
  , if_quot^#(true(), x, y) ->
    c_7(quot^#(minus(x, y), y), minus^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
  , if_quot(true(), x, y) -> s(quot(minus(x, y), y))
  , if_quot(false(), x, y) -> 0() }
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)
    , le(0(), y) -> true()
    , le(s(x), 0()) -> false()
    , le(s(x), s(y)) -> le(x, y) }

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))
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(x, s(y)) ->
    c_6(if_quot^#(le(s(y), x), x, s(y)), le^#(s(y), x))
  , if_quot^#(true(), x, y) ->
    c_7(quot^#(minus(x, y), y), minus^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..