MAYBE

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

Strict Trs:
  { minus(x, y) -> cond(equal(min(x, y), y), x, y)
  , cond(true(), x, y) -> s(minus(x, s(y)))
  , equal(s(x), s(y)) -> equal(x, y)
  , equal(s(x), 0()) -> false()
  , equal(0(), s(y)) -> false()
  , equal(0(), 0()) -> true()
  , min(u, 0()) -> 0()
  , min(s(u), s(v)) -> s(min(u, v))
  , min(0(), v) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { minus^#(x, y) ->
    c_1(cond^#(equal(min(x, y), y), x, y),
        equal^#(min(x, y), y),
        min^#(x, y))
  , cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
  , equal^#(s(x), s(y)) -> c_3(equal^#(x, y))
  , equal^#(s(x), 0()) -> c_4()
  , equal^#(0(), s(y)) -> c_5()
  , equal^#(0(), 0()) -> c_6()
  , min^#(u, 0()) -> c_7()
  , min^#(s(u), s(v)) -> c_8(min^#(u, v))
  , min^#(0(), v) -> c_9() }

and mark the set of starting terms.

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

Strict DPs:
  { minus^#(x, y) ->
    c_1(cond^#(equal(min(x, y), y), x, y),
        equal^#(min(x, y), y),
        min^#(x, y))
  , cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
  , equal^#(s(x), s(y)) -> c_3(equal^#(x, y))
  , equal^#(s(x), 0()) -> c_4()
  , equal^#(0(), s(y)) -> c_5()
  , equal^#(0(), 0()) -> c_6()
  , min^#(u, 0()) -> c_7()
  , min^#(s(u), s(v)) -> c_8(min^#(u, v))
  , min^#(0(), v) -> c_9() }
Weak Trs:
  { minus(x, y) -> cond(equal(min(x, y), y), x, y)
  , cond(true(), x, y) -> s(minus(x, s(y)))
  , equal(s(x), s(y)) -> equal(x, y)
  , equal(s(x), 0()) -> false()
  , equal(0(), s(y)) -> false()
  , equal(0(), 0()) -> true()
  , min(u, 0()) -> 0()
  , min(s(u), s(v)) -> s(min(u, v))
  , min(0(), v) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: minus^#(x, y) ->
         c_1(cond^#(equal(min(x, y), y), x, y),
             equal^#(min(x, y), y),
             min^#(x, y))
    , 2: cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
    , 3: equal^#(s(x), s(y)) -> c_3(equal^#(x, y))
    , 4: equal^#(s(x), 0()) -> c_4()
    , 5: equal^#(0(), s(y)) -> c_5()
    , 6: equal^#(0(), 0()) -> c_6()
    , 7: min^#(u, 0()) -> c_7()
    , 8: min^#(s(u), s(v)) -> c_8(min^#(u, v))
    , 9: min^#(0(), v) -> c_9() }

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

Strict DPs:
  { minus^#(x, y) ->
    c_1(cond^#(equal(min(x, y), y), x, y),
        equal^#(min(x, y), y),
        min^#(x, y))
  , cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
  , equal^#(s(x), s(y)) -> c_3(equal^#(x, y))
  , min^#(s(u), s(v)) -> c_8(min^#(u, v)) }
Weak DPs:
  { equal^#(s(x), 0()) -> c_4()
  , equal^#(0(), s(y)) -> c_5()
  , equal^#(0(), 0()) -> c_6()
  , min^#(u, 0()) -> c_7()
  , min^#(0(), v) -> c_9() }
Weak Trs:
  { minus(x, y) -> cond(equal(min(x, y), y), x, y)
  , cond(true(), x, y) -> s(minus(x, s(y)))
  , equal(s(x), s(y)) -> equal(x, y)
  , equal(s(x), 0()) -> false()
  , equal(0(), s(y)) -> false()
  , equal(0(), 0()) -> true()
  , min(u, 0()) -> 0()
  , min(s(u), s(v)) -> s(min(u, v))
  , min(0(), v) -> 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.

{ equal^#(s(x), 0()) -> c_4()
, equal^#(0(), s(y)) -> c_5()
, equal^#(0(), 0()) -> c_6()
, min^#(u, 0()) -> c_7()
, min^#(0(), v) -> c_9() }

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

Strict DPs:
  { minus^#(x, y) ->
    c_1(cond^#(equal(min(x, y), y), x, y),
        equal^#(min(x, y), y),
        min^#(x, y))
  , cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
  , equal^#(s(x), s(y)) -> c_3(equal^#(x, y))
  , min^#(s(u), s(v)) -> c_8(min^#(u, v)) }
Weak Trs:
  { minus(x, y) -> cond(equal(min(x, y), y), x, y)
  , cond(true(), x, y) -> s(minus(x, s(y)))
  , equal(s(x), s(y)) -> equal(x, y)
  , equal(s(x), 0()) -> false()
  , equal(0(), s(y)) -> false()
  , equal(0(), 0()) -> true()
  , min(u, 0()) -> 0()
  , min(s(u), s(v)) -> s(min(u, v))
  , min(0(), v) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { equal(s(x), s(y)) -> equal(x, y)
    , equal(s(x), 0()) -> false()
    , equal(0(), s(y)) -> false()
    , equal(0(), 0()) -> true()
    , min(u, 0()) -> 0()
    , min(s(u), s(v)) -> s(min(u, v))
    , min(0(), v) -> 0() }

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

Strict DPs:
  { minus^#(x, y) ->
    c_1(cond^#(equal(min(x, y), y), x, y),
        equal^#(min(x, y), y),
        min^#(x, y))
  , cond^#(true(), x, y) -> c_2(minus^#(x, s(y)))
  , equal^#(s(x), s(y)) -> c_3(equal^#(x, y))
  , min^#(s(u), s(v)) -> c_8(min^#(u, v)) }
Weak Trs:
  { equal(s(x), s(y)) -> equal(x, y)
  , equal(s(x), 0()) -> false()
  , equal(0(), s(y)) -> false()
  , equal(0(), 0()) -> true()
  , min(u, 0()) -> 0()
  , min(s(u), s(v)) -> s(min(u, v))
  , min(0(), v) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..