MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            id_inc(x) -> x
            id_inc(x) -> s(x)
            if(false(),x,y) -> y
            if(true(),x,y) -> rand(p(x),id_inc(y))
            nonZero(0()) -> false()
            nonZero(s(x)) -> true()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            rand(x,y) -> if(nonZero(x),x,y)
            random(x) -> rand(x,0())
        - Signature:
            {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id_inc,if,nonZero,p,rand,random} and constructors {0
            ,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          id_inc#(x) -> c_1()
          id_inc#(x) -> c_2()
          if#(false(),x,y) -> c_3()
          if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
          nonZero#(0()) -> c_5()
          nonZero#(s(x)) -> c_6()
          p#(s(0())) -> c_7()
          p#(s(s(x))) -> c_8(p#(s(x)))
          rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
          random#(x) -> c_10(rand#(x,0()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            id_inc#(x) -> c_1()
            id_inc#(x) -> c_2()
            if#(false(),x,y) -> c_3()
            if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
            nonZero#(0()) -> c_5()
            nonZero#(s(x)) -> c_6()
            p#(s(0())) -> c_7()
            p#(s(s(x))) -> c_8(p#(s(x)))
            rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
            random#(x) -> c_10(rand#(x,0()))
        - Weak TRS:
            id_inc(x) -> x
            id_inc(x) -> s(x)
            if(false(),x,y) -> y
            if(true(),x,y) -> rand(p(x),id_inc(y))
            nonZero(0()) -> false()
            nonZero(s(x)) -> true()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            rand(x,y) -> if(nonZero(x),x,y)
            random(x) -> rand(x,0())
        - Signature:
            {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1,id_inc#/1,if#/3,nonZero#/1,p#/1,rand#/2,random#/1} / {0/0
            ,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/3,c_5/0,c_6/0,c_7/0,c_8/1,c_9/2,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id_inc#,if#,nonZero#,p#,rand#
            ,random#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          id_inc(x) -> x
          id_inc(x) -> s(x)
          nonZero(0()) -> false()
          nonZero(s(x)) -> true()
          p(s(0())) -> 0()
          p(s(s(x))) -> s(p(s(x)))
          id_inc#(x) -> c_1()
          id_inc#(x) -> c_2()
          if#(false(),x,y) -> c_3()
          if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
          nonZero#(0()) -> c_5()
          nonZero#(s(x)) -> c_6()
          p#(s(0())) -> c_7()
          p#(s(s(x))) -> c_8(p#(s(x)))
          rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
          random#(x) -> c_10(rand#(x,0()))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            id_inc#(x) -> c_1()
            id_inc#(x) -> c_2()
            if#(false(),x,y) -> c_3()
            if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
            nonZero#(0()) -> c_5()
            nonZero#(s(x)) -> c_6()
            p#(s(0())) -> c_7()
            p#(s(s(x))) -> c_8(p#(s(x)))
            rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
            random#(x) -> c_10(rand#(x,0()))
        - Weak TRS:
            id_inc(x) -> x
            id_inc(x) -> s(x)
            nonZero(0()) -> false()
            nonZero(s(x)) -> true()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1,id_inc#/1,if#/3,nonZero#/1,p#/1,rand#/2,random#/1} / {0/0
            ,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/3,c_5/0,c_6/0,c_7/0,c_8/1,c_9/2,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id_inc#,if#,nonZero#,p#,rand#
            ,random#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,5,6,7}
        by application of
          Pre({1,2,3,5,6,7}) = {4,8,9}.
        Here rules are labelled as follows:
          1: id_inc#(x) -> c_1()
          2: id_inc#(x) -> c_2()
          3: if#(false(),x,y) -> c_3()
          4: if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
          5: nonZero#(0()) -> c_5()
          6: nonZero#(s(x)) -> c_6()
          7: p#(s(0())) -> c_7()
          8: p#(s(s(x))) -> c_8(p#(s(x)))
          9: rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
          10: random#(x) -> c_10(rand#(x,0()))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
            p#(s(s(x))) -> c_8(p#(s(x)))
            rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
            random#(x) -> c_10(rand#(x,0()))
        - Weak DPs:
            id_inc#(x) -> c_1()
            id_inc#(x) -> c_2()
            if#(false(),x,y) -> c_3()
            nonZero#(0()) -> c_5()
            nonZero#(s(x)) -> c_6()
            p#(s(0())) -> c_7()
        - Weak TRS:
            id_inc(x) -> x
            id_inc(x) -> s(x)
            nonZero(0()) -> false()
            nonZero(s(x)) -> true()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1,id_inc#/1,if#/3,nonZero#/1,p#/1,rand#/2,random#/1} / {0/0
            ,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/3,c_5/0,c_6/0,c_7/0,c_8/1,c_9/2,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id_inc#,if#,nonZero#,p#,rand#
            ,random#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
             -->_1 rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x)):3
             -->_2 p#(s(s(x))) -> c_8(p#(s(x))):2
             -->_2 p#(s(0())) -> c_7():10
             -->_3 id_inc#(x) -> c_2():6
             -->_3 id_inc#(x) -> c_1():5
          
          2:S:p#(s(s(x))) -> c_8(p#(s(x)))
             -->_1 p#(s(0())) -> c_7():10
             -->_1 p#(s(s(x))) -> c_8(p#(s(x))):2
          
          3:S:rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
             -->_2 nonZero#(s(x)) -> c_6():9
             -->_2 nonZero#(0()) -> c_5():8
             -->_1 if#(false(),x,y) -> c_3():7
             -->_1 if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y)):1
          
          4:S:random#(x) -> c_10(rand#(x,0()))
             -->_1 rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x)):3
          
          5:W:id_inc#(x) -> c_1()
             
          
          6:W:id_inc#(x) -> c_2()
             
          
          7:W:if#(false(),x,y) -> c_3()
             
          
          8:W:nonZero#(0()) -> c_5()
             
          
          9:W:nonZero#(s(x)) -> c_6()
             
          
          10:W:p#(s(0())) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: id_inc#(x) -> c_1()
          6: id_inc#(x) -> c_2()
          10: p#(s(0())) -> c_7()
          7: if#(false(),x,y) -> c_3()
          8: nonZero#(0()) -> c_5()
          9: nonZero#(s(x)) -> c_6()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
            p#(s(s(x))) -> c_8(p#(s(x)))
            rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
            random#(x) -> c_10(rand#(x,0()))
        - Weak TRS:
            id_inc(x) -> x
            id_inc(x) -> s(x)
            nonZero(0()) -> false()
            nonZero(s(x)) -> true()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1,id_inc#/1,if#/3,nonZero#/1,p#/1,rand#/2,random#/1} / {0/0
            ,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/3,c_5/0,c_6/0,c_7/0,c_8/1,c_9/2,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id_inc#,if#,nonZero#,p#,rand#
            ,random#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y))
             -->_1 rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x)):3
             -->_2 p#(s(s(x))) -> c_8(p#(s(x))):2
          
          2:S:p#(s(s(x))) -> c_8(p#(s(x)))
             -->_1 p#(s(s(x))) -> c_8(p#(s(x))):2
          
          3:S:rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x))
             -->_1 if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x),id_inc#(y)):1
          
          4:S:random#(x) -> c_10(rand#(x,0()))
             -->_1 rand#(x,y) -> c_9(if#(nonZero(x),x,y),nonZero#(x)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x))
          rand#(x,y) -> c_9(if#(nonZero(x),x,y))
* Step 6: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x))
            p#(s(s(x))) -> c_8(p#(s(x)))
            rand#(x,y) -> c_9(if#(nonZero(x),x,y))
            random#(x) -> c_10(rand#(x,0()))
        - Weak TRS:
            id_inc(x) -> x
            id_inc(x) -> s(x)
            nonZero(0()) -> false()
            nonZero(s(x)) -> true()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
        - Signature:
            {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1,id_inc#/1,if#/3,nonZero#/1,p#/1,rand#/2,random#/1} / {0/0
            ,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id_inc#,if#,nonZero#,p#,rand#
            ,random#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x))
           -->_1 rand#(x,y) -> c_9(if#(nonZero(x),x,y)):3
           -->_2 p#(s(s(x))) -> c_8(p#(s(x))):2
        
        2:S:p#(s(s(x))) -> c_8(p#(s(x)))
           -->_1 p#(s(s(x))) -> c_8(p#(s(x))):2
        
        3:S:rand#(x,y) -> c_9(if#(nonZero(x),x,y))
           -->_1 if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x)):1
        
        4:S:random#(x) -> c_10(rand#(x,0()))
           -->_1 rand#(x,y) -> c_9(if#(nonZero(x),x,y)):3
        
        
        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).
        
        [(4,random#(x) -> c_10(rand#(x,0())))]
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          if#(true(),x,y) -> c_4(rand#(p(x),id_inc(y)),p#(x))
          p#(s(s(x))) -> c_8(p#(s(x)))
          rand#(x,y) -> c_9(if#(nonZero(x),x,y))
      - Weak TRS:
          id_inc(x) -> x
          id_inc(x) -> s(x)
          nonZero(0()) -> false()
          nonZero(s(x)) -> true()
          p(s(0())) -> 0()
          p(s(s(x))) -> s(p(s(x)))
      - Signature:
          {id_inc/1,if/3,nonZero/1,p/1,rand/2,random/1,id_inc#/1,if#/3,nonZero#/1,p#/1,rand#/2,random#/1} / {0/0
          ,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1,c_10/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {id_inc#,if#,nonZero#,p#,rand#
          ,random#} and constructors {0,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE