MAYBE

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

Strict Trs:
  { diff(x, y) -> cond1(equal(x, y), x, y)
  , cond1(true(), x, y) -> 0()
  , cond1(false(), x, y) -> cond2(gt(x, y), x, y)
  , equal(0(), 0()) -> true()
  , equal(0(), s(y)) -> false()
  , equal(s(x), 0()) -> false()
  , equal(s(x), s(y)) -> equal(x, y)
  , cond2(true(), x, y) -> s(diff(x, s(y)))
  , cond2(false(), x, y) -> s(diff(s(x), y))
  , gt(0(), v) -> false()
  , gt(s(u), 0()) -> true()
  , gt(s(u), s(v)) -> gt(u, v) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y), equal^#(x, y))
  , cond1^#(true(), x, y) -> c_2()
  , cond1^#(false(), x, y) ->
    c_3(cond2^#(gt(x, y), x, y), gt^#(x, y))
  , equal^#(0(), 0()) -> c_4()
  , equal^#(0(), s(y)) -> c_5()
  , equal^#(s(x), 0()) -> c_6()
  , equal^#(s(x), s(y)) -> c_7(equal^#(x, y))
  , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y)))
  , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y))
  , gt^#(0(), v) -> c_10()
  , gt^#(s(u), 0()) -> c_11()
  , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) }

and mark the set of starting terms.

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

Strict DPs:
  { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y), equal^#(x, y))
  , cond1^#(true(), x, y) -> c_2()
  , cond1^#(false(), x, y) ->
    c_3(cond2^#(gt(x, y), x, y), gt^#(x, y))
  , equal^#(0(), 0()) -> c_4()
  , equal^#(0(), s(y)) -> c_5()
  , equal^#(s(x), 0()) -> c_6()
  , equal^#(s(x), s(y)) -> c_7(equal^#(x, y))
  , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y)))
  , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y))
  , gt^#(0(), v) -> c_10()
  , gt^#(s(u), 0()) -> c_11()
  , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) }
Weak Trs:
  { diff(x, y) -> cond1(equal(x, y), x, y)
  , cond1(true(), x, y) -> 0()
  , cond1(false(), x, y) -> cond2(gt(x, y), x, y)
  , equal(0(), 0()) -> true()
  , equal(0(), s(y)) -> false()
  , equal(s(x), 0()) -> false()
  , equal(s(x), s(y)) -> equal(x, y)
  , cond2(true(), x, y) -> s(diff(x, s(y)))
  , cond2(false(), x, y) -> s(diff(s(x), y))
  , gt(0(), v) -> false()
  , gt(s(u), 0()) -> true()
  , gt(s(u), s(v)) -> gt(u, v) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2,4,5,6,10,11} by
applications of Pre({2,4,5,6,10,11}) = {1,3,7,12}. Here rules are
labeled as follows:

  DPs:
    { 1: diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y), equal^#(x, y))
    , 2: cond1^#(true(), x, y) -> c_2()
    , 3: cond1^#(false(), x, y) ->
         c_3(cond2^#(gt(x, y), x, y), gt^#(x, y))
    , 4: equal^#(0(), 0()) -> c_4()
    , 5: equal^#(0(), s(y)) -> c_5()
    , 6: equal^#(s(x), 0()) -> c_6()
    , 7: equal^#(s(x), s(y)) -> c_7(equal^#(x, y))
    , 8: cond2^#(true(), x, y) -> c_8(diff^#(x, s(y)))
    , 9: cond2^#(false(), x, y) -> c_9(diff^#(s(x), y))
    , 10: gt^#(0(), v) -> c_10()
    , 11: gt^#(s(u), 0()) -> c_11()
    , 12: gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) }

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

Strict DPs:
  { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y), equal^#(x, y))
  , cond1^#(false(), x, y) ->
    c_3(cond2^#(gt(x, y), x, y), gt^#(x, y))
  , equal^#(s(x), s(y)) -> c_7(equal^#(x, y))
  , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y)))
  , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y))
  , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) }
Weak DPs:
  { cond1^#(true(), x, y) -> c_2()
  , equal^#(0(), 0()) -> c_4()
  , equal^#(0(), s(y)) -> c_5()
  , equal^#(s(x), 0()) -> c_6()
  , gt^#(0(), v) -> c_10()
  , gt^#(s(u), 0()) -> c_11() }
Weak Trs:
  { diff(x, y) -> cond1(equal(x, y), x, y)
  , cond1(true(), x, y) -> 0()
  , cond1(false(), x, y) -> cond2(gt(x, y), x, y)
  , equal(0(), 0()) -> true()
  , equal(0(), s(y)) -> false()
  , equal(s(x), 0()) -> false()
  , equal(s(x), s(y)) -> equal(x, y)
  , cond2(true(), x, y) -> s(diff(x, s(y)))
  , cond2(false(), x, y) -> s(diff(s(x), y))
  , gt(0(), v) -> false()
  , gt(s(u), 0()) -> true()
  , gt(s(u), s(v)) -> gt(u, v) }
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.

{ cond1^#(true(), x, y) -> c_2()
, equal^#(0(), 0()) -> c_4()
, equal^#(0(), s(y)) -> c_5()
, equal^#(s(x), 0()) -> c_6()
, gt^#(0(), v) -> c_10()
, gt^#(s(u), 0()) -> c_11() }

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

Strict DPs:
  { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y), equal^#(x, y))
  , cond1^#(false(), x, y) ->
    c_3(cond2^#(gt(x, y), x, y), gt^#(x, y))
  , equal^#(s(x), s(y)) -> c_7(equal^#(x, y))
  , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y)))
  , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y))
  , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) }
Weak Trs:
  { diff(x, y) -> cond1(equal(x, y), x, y)
  , cond1(true(), x, y) -> 0()
  , cond1(false(), x, y) -> cond2(gt(x, y), x, y)
  , equal(0(), 0()) -> true()
  , equal(0(), s(y)) -> false()
  , equal(s(x), 0()) -> false()
  , equal(s(x), s(y)) -> equal(x, y)
  , cond2(true(), x, y) -> s(diff(x, s(y)))
  , cond2(false(), x, y) -> s(diff(s(x), y))
  , gt(0(), v) -> false()
  , gt(s(u), 0()) -> true()
  , gt(s(u), s(v)) -> gt(u, v) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y), equal^#(x, y))
  , cond1^#(false(), x, y) ->
    c_3(cond2^#(gt(x, y), x, y), gt^#(x, y))
  , equal^#(s(x), s(y)) -> c_7(equal^#(x, y))
  , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y)))
  , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y))
  , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) }
Weak Trs:
  { equal(0(), 0()) -> true()
  , equal(0(), s(y)) -> false()
  , equal(s(x), 0()) -> false()
  , equal(s(x), s(y)) -> equal(x, y)
  , gt(0(), v) -> false()
  , gt(s(u), 0()) -> true()
  , gt(s(u), s(v)) -> gt(u, v) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..