MAYBE

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

Strict Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(0()) -> 0()
  , p(s(x)) -> 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 weak dependency pairs:

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

  Strict Usable Rules:
    { nonZero(0()) -> false()
    , nonZero(s(x)) -> true()
    , p(0()) -> 0()
    , p(s(x)) -> 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:
  { nonZero^#(0()) -> c_1()
  , nonZero^#(s(x)) -> c_2()
  , p^#(0()) -> c_3()
  , p^#(s(x)) -> c_4()
  , 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))
  , if^#(false(), x, y) -> c_9()
  , if^#(true(), x, y) -> c_10(rand^#(p(x), id_inc(y))) }
Strict Trs:
  { nonZero(0()) -> false()
  , nonZero(s(x)) -> true()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , id_inc(x) -> x
  , id_inc(x) -> s(x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_7) = {1}, Uargs(rand^#) = {1, 2}, Uargs(c_8) = {1},
  Uargs(if^#) = {1}, Uargs(c_10) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

       [nonZero](x1) = [1]                           
                                                     
                 [0] = [2]                           
                                                     
             [false] = [0]                           
                                                     
             [s](x1) = [1] x1 + [0]                  
                                                     
              [true] = [0]                           
                                                     
             [p](x1) = [1] x1 + [2]                  
                                                     
        [id_inc](x1) = [1] x1 + [2]                  
                                                     
     [nonZero^#](x1) = [2] x1 + [2]                  
                                                     
               [c_1] = [1]                           
                                                     
               [c_2] = [1]                           
                                                     
           [p^#](x1) = [1] x1 + [2]                  
                                                     
               [c_3] = [1]                           
                                                     
               [c_4] = [1]                           
                                                     
      [id_inc^#](x1) = [1]                           
                                                     
               [c_5] = [2]                           
                                                     
               [c_6] = [2]                           
                                                     
      [random^#](x1) = [1] x1 + [1]                  
                                                     
           [c_7](x1) = [1] x1 + [1]                  
                                                     
    [rand^#](x1, x2) = [1] x1 + [2] x2 + [1]         
                                                     
           [c_8](x1) = [1] x1 + [2]                  
                                                     
  [if^#](x1, x2, x3) = [1] x1 + [1] x2 + [2] x3 + [1]
                                                     
               [c_9] = [0]                           
                                                     
          [c_10](x1) = [1] x1 + [0]                  

This order satisfies following ordering constraints:

   [nonZero(0())] = [1]        
                  > [0]        
                  = [false()]  
                               
  [nonZero(s(x))] = [1]        
                  > [0]        
                  = [true()]   
                               
         [p(0())] = [4]        
                  > [2]        
                  = [0()]      
                               
        [p(s(x))] = [1] x + [2]
                  > [1] x + [0]
                  = [x]        
                               
      [id_inc(x)] = [1] x + [2]
                  > [1] x + [0]
                  = [x]        
                               
      [id_inc(x)] = [1] x + [2]
                  > [1] x + [0]
                  = [s(x)]     
                               

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

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

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

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

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

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

Consider the dependency graph

  1: random^#(x) -> c_7(rand^#(x, 0()))
     -->_1 rand^#(x, y) -> c_8(if^#(nonZero(x), x, y)) :2
  
  2: rand^#(x, y) -> c_8(if^#(nonZero(x), x, y))
     -->_1 if^#(true(), x, y) -> c_10(rand^#(p(x), id_inc(y))) :3
  
  3: if^#(true(), x, y) -> c_10(rand^#(p(x), id_inc(y)))
     -->_1 rand^#(x, y) -> c_8(if^#(nonZero(x), x, y)) :2
  

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_7(rand^#(x, 0())) }


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

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

The input cannot be shown compatible

Arrrr..