MAYBE

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

Strict Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , random(x) -> rand(x, 0())
  , rand(x, y) -> if(nonZero(x), x, y)
  , if(false(), x, y) -> y
  , if(true(), x, y) -> rand(p(x), id_inc(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { nonZero^#(0()) -> c_1()
  , nonZero^#(s(x)) -> c_2()
  , p^#(s(0())) -> c_3()
  , p^#(s(s(x))) -> c_4(p^#(s(x)))
  , id_inc^#(x) -> c_5()
  , id_inc^#(x) -> c_6()
  , random^#(x) -> c_7(rand^#(x, 0()))
  , rand^#(x, y) -> c_8(if^#(nonZero(x), x, y), nonZero^#(x))
  , if^#(false(), x, y) -> c_9()
  , if^#(true(), x, y) ->
    c_10(rand^#(p(x), id_inc(y)), p^#(x), id_inc^#(y)) }

and mark the set of starting terms.

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

Strict DPs:
  { nonZero^#(0()) -> c_1()
  , nonZero^#(s(x)) -> c_2()
  , p^#(s(0())) -> c_3()
  , p^#(s(s(x))) -> c_4(p^#(s(x)))
  , id_inc^#(x) -> c_5()
  , id_inc^#(x) -> c_6()
  , random^#(x) -> c_7(rand^#(x, 0()))
  , rand^#(x, y) -> c_8(if^#(nonZero(x), x, y), nonZero^#(x))
  , if^#(false(), x, y) -> c_9()
  , if^#(true(), x, y) ->
    c_10(rand^#(p(x), id_inc(y)), p^#(x), id_inc^#(y)) }
Weak Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , random(x) -> rand(x, 0())
  , rand(x, y) -> if(nonZero(x), x, y)
  , if(false(), x, y) -> y
  , if(true(), x, y) -> rand(p(x), id_inc(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,5,6,9} by
applications of Pre({1,2,3,5,6,9}) = {4,8,10}. Here rules are
labeled as follows:

  DPs:
    { 1: nonZero^#(0()) -> c_1()
    , 2: nonZero^#(s(x)) -> c_2()
    , 3: p^#(s(0())) -> c_3()
    , 4: p^#(s(s(x))) -> c_4(p^#(s(x)))
    , 5: id_inc^#(x) -> c_5()
    , 6: id_inc^#(x) -> c_6()
    , 7: random^#(x) -> c_7(rand^#(x, 0()))
    , 8: rand^#(x, y) -> c_8(if^#(nonZero(x), x, y), nonZero^#(x))
    , 9: if^#(false(), x, y) -> c_9()
    , 10: if^#(true(), x, y) ->
          c_10(rand^#(p(x), id_inc(y)), p^#(x), id_inc^#(y)) }

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

Strict DPs:
  { p^#(s(s(x))) -> c_4(p^#(s(x)))
  , random^#(x) -> c_7(rand^#(x, 0()))
  , rand^#(x, y) -> c_8(if^#(nonZero(x), x, y), nonZero^#(x))
  , if^#(true(), x, y) ->
    c_10(rand^#(p(x), id_inc(y)), p^#(x), id_inc^#(y)) }
Weak DPs:
  { nonZero^#(0()) -> c_1()
  , nonZero^#(s(x)) -> c_2()
  , p^#(s(0())) -> c_3()
  , id_inc^#(x) -> c_5()
  , id_inc^#(x) -> c_6()
  , if^#(false(), x, y) -> c_9() }
Weak Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , random(x) -> rand(x, 0())
  , rand(x, y) -> if(nonZero(x), x, y)
  , if(false(), x, y) -> y
  , if(true(), x, y) -> rand(p(x), id_inc(y)) }
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.

{ nonZero^#(0()) -> c_1()
, nonZero^#(s(x)) -> c_2()
, p^#(s(0())) -> c_3()
, id_inc^#(x) -> c_5()
, id_inc^#(x) -> c_6()
, if^#(false(), x, y) -> c_9() }

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

Strict DPs:
  { p^#(s(s(x))) -> c_4(p^#(s(x)))
  , random^#(x) -> c_7(rand^#(x, 0()))
  , rand^#(x, y) -> c_8(if^#(nonZero(x), x, y), nonZero^#(x))
  , if^#(true(), x, y) ->
    c_10(rand^#(p(x), id_inc(y)), p^#(x), id_inc^#(y)) }
Weak Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , random(x) -> rand(x, 0())
  , rand(x, y) -> if(nonZero(x), x, y)
  , if(false(), x, y) -> y
  , if(true(), x, y) -> rand(p(x), id_inc(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { rand^#(x, y) -> c_8(if^#(nonZero(x), x, y), nonZero^#(x))
  , if^#(true(), x, y) ->
    c_10(rand^#(p(x), id_inc(y)), p^#(x), id_inc^#(y)) }

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

Strict DPs:
  { p^#(s(s(x))) -> c_1(p^#(s(x)))
  , random^#(x) -> c_2(rand^#(x, 0()))
  , rand^#(x, y) -> c_3(if^#(nonZero(x), x, y))
  , if^#(true(), x, y) -> c_4(rand^#(p(x), id_inc(y)), p^#(x)) }
Weak Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x)
  , random(x) -> rand(x, 0())
  , rand(x, y) -> if(nonZero(x), x, y)
  , if(false(), x, y) -> y
  , if(true(), x, y) -> rand(p(x), id_inc(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { nonZero(0()) -> false()
    , nonZero(s(x)) -> true()
    , p(s(0())) -> 0()
    , p(s(s(x))) -> s(p(s(x)))
    , id_inc(x) -> x
    , id_inc(x) -> s(x) }

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

Strict DPs:
  { p^#(s(s(x))) -> c_1(p^#(s(x)))
  , random^#(x) -> c_2(rand^#(x, 0()))
  , rand^#(x, y) -> c_3(if^#(nonZero(x), x, y))
  , if^#(true(), x, y) -> c_4(rand^#(p(x), id_inc(y)), p^#(x)) }
Weak Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: p^#(s(s(x))) -> c_1(p^#(s(x)))
     -->_1 p^#(s(s(x))) -> c_1(p^#(s(x))) :1
  
  2: random^#(x) -> c_2(rand^#(x, 0()))
     -->_1 rand^#(x, y) -> c_3(if^#(nonZero(x), x, y)) :3
  
  3: rand^#(x, y) -> c_3(if^#(nonZero(x), x, y))
     -->_1 if^#(true(), x, y) -> c_4(rand^#(p(x), id_inc(y)), p^#(x)) :4
  
  4: if^#(true(), x, y) -> c_4(rand^#(p(x), id_inc(y)), p^#(x))
     -->_1 rand^#(x, y) -> c_3(if^#(nonZero(x), x, y)) :3
     -->_2 p^#(s(s(x))) -> c_1(p^#(s(x))) :1
  

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).

  { random^#(x) -> c_2(rand^#(x, 0())) }


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

Strict DPs:
  { p^#(s(s(x))) -> c_1(p^#(s(x)))
  , rand^#(x, y) -> c_3(if^#(nonZero(x), x, y))
  , if^#(true(), x, y) -> c_4(rand^#(p(x), id_inc(y)), p^#(x)) }
Weak Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(s(0())) -> 0()
  , p(s(s(x))) -> s(p(s(x)))
  , id_inc(x) -> x
  , id_inc(x) -> s(x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..