YES(O(1),O(1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Strict Trs:
  { f(x, y, z) -> g(<=(x, y), x, y, z)
  , g(true(), x, y, z) -> z
  , g(false(), x, y, z) ->
    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following dependency tuples:

Strict DPs:
  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
  , g^#(true(), x, y, z) -> c_2()
  , g^#(false(), x, y, z) ->
    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
        f^#(p(x), y, z),
        p^#(x),
        f^#(p(y), z, x),
        p^#(y),
        f^#(p(z), x, y),
        p^#(z))
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Strict DPs:
  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
  , g^#(true(), x, y, z) -> c_2()
  , g^#(false(), x, y, z) ->
    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
        f^#(p(x), y, z),
        p^#(x),
        f^#(p(y), z, x),
        p^#(y),
        f^#(p(z), x, y),
        p^#(z))
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Weak Trs:
  { f(x, y, z) -> g(<=(x, y), x, y, z)
  , g(true(), x, y, z) -> z
  , g(false(), x, y, z) ->
    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
    , 2: g^#(true(), x, y, z) -> c_2()
    , 3: g^#(false(), x, y, z) ->
         c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
             f^#(p(x), y, z),
             p^#(x),
             f^#(p(y), z, x),
             p^#(y),
             f^#(p(z), x, y),
             p^#(z))
    , 4: p^#(0()) -> c_4()
    , 5: p^#(s(x)) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Strict DPs:
  { g^#(false(), x, y, z) ->
    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
        f^#(p(x), y, z),
        p^#(x),
        f^#(p(y), z, x),
        p^#(y),
        f^#(p(z), x, y),
        p^#(z)) }
Weak DPs:
  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
  , g^#(true(), x, y, z) -> c_2()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Weak Trs:
  { f(x, y, z) -> g(<=(x, y), x, y, z)
  , g(true(), x, y, z) -> z
  , g(false(), x, y, z) ->
    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {1} by applications of
Pre({1}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: g^#(false(), x, y, z) ->
         c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
             f^#(p(x), y, z),
             p^#(x),
             f^#(p(y), z, x),
             p^#(y),
             f^#(p(z), x, y),
             p^#(z))
    , 2: f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
    , 3: g^#(true(), x, y, z) -> c_2()
    , 4: p^#(0()) -> c_4()
    , 5: p^#(s(x)) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak DPs:
  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
  , g^#(true(), x, y, z) -> c_2()
  , g^#(false(), x, y, z) ->
    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
        f^#(p(x), y, z),
        p^#(x),
        f^#(p(y), z, x),
        p^#(y),
        f^#(p(z), x, y),
        p^#(z))
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Weak Trs:
  { f(x, y, z) -> g(<=(x, y), x, y, z)
  , g(true(), x, y, z) -> z
  , g(false(), x, y, z) ->
    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
, g^#(true(), x, y, z) -> c_2()
, g^#(false(), x, y, z) ->
  c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
      f^#(p(x), y, z),
      p^#(x),
      f^#(p(y), z, x),
      p^#(y),
      f^#(p(z), x, y),
      p^#(z))
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak Trs:
  { f(x, y, z) -> g(<=(x, y), x, y, z)
  , g(true(), x, y, z) -> z
  , g(false(), x, y, z) ->
    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(1))