MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
            zero(cons(x,xs)) -> zero2(sub(x,x),cons(x,xs))
            zero(nil()) -> zero2(0(),nil())
            zero2(0(),cons(x,xs)) -> cons(sub(x,x),zero(xs))
            zero2(0(),nil()) -> nil()
            zero2(s(y),cons(x,xs)) -> zero(cons(x,xs))
            zero2(s(y),nil()) -> zero(nil())
        - Signature:
            {sub/2,zero/1,zero2/2} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub,zero,zero2} and constructors {0,cons,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          sub#(0(),0()) -> c_1()
          sub#(0(),s(x)) -> c_2()
          sub#(s(x),0()) -> c_3()
          sub#(s(x),s(y)) -> c_4(sub#(x,y))
          zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
          zero#(nil()) -> c_6(zero2#(0(),nil()))
          zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
          zero2#(0(),nil()) -> c_8()
          zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
          zero2#(s(y),nil()) -> c_10(zero#(nil()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            sub#(0(),0()) -> c_1()
            sub#(0(),s(x)) -> c_2()
            sub#(s(x),0()) -> c_3()
            sub#(s(x),s(y)) -> c_4(sub#(x,y))
            zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
            zero#(nil()) -> c_6(zero2#(0(),nil()))
            zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
            zero2#(0(),nil()) -> c_8()
            zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
            zero2#(s(y),nil()) -> c_10(zero#(nil()))
        - Weak TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
            zero(cons(x,xs)) -> zero2(sub(x,x),cons(x,xs))
            zero(nil()) -> zero2(0(),nil())
            zero2(0(),cons(x,xs)) -> cons(sub(x,x),zero(xs))
            zero2(0(),nil()) -> nil()
            zero2(s(y),cons(x,xs)) -> zero(cons(x,xs))
            zero2(s(y),nil()) -> zero(nil())
        - Signature:
            {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
            ,c_7/2,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sub(0(),0()) -> 0()
          sub(0(),s(x)) -> 0()
          sub(s(x),0()) -> s(x)
          sub(s(x),s(y)) -> sub(x,y)
          sub#(0(),0()) -> c_1()
          sub#(0(),s(x)) -> c_2()
          sub#(s(x),0()) -> c_3()
          sub#(s(x),s(y)) -> c_4(sub#(x,y))
          zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
          zero#(nil()) -> c_6(zero2#(0(),nil()))
          zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
          zero2#(0(),nil()) -> c_8()
          zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
          zero2#(s(y),nil()) -> c_10(zero#(nil()))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            sub#(0(),0()) -> c_1()
            sub#(0(),s(x)) -> c_2()
            sub#(s(x),0()) -> c_3()
            sub#(s(x),s(y)) -> c_4(sub#(x,y))
            zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
            zero#(nil()) -> c_6(zero2#(0(),nil()))
            zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
            zero2#(0(),nil()) -> c_8()
            zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
            zero2#(s(y),nil()) -> c_10(zero#(nil()))
        - Weak TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
        - Signature:
            {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
            ,c_7/2,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,8}
        by application of
          Pre({1,2,3,8}) = {4,5,6,7}.
        Here rules are labelled as follows:
          1: sub#(0(),0()) -> c_1()
          2: sub#(0(),s(x)) -> c_2()
          3: sub#(s(x),0()) -> c_3()
          4: sub#(s(x),s(y)) -> c_4(sub#(x,y))
          5: zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
          6: zero#(nil()) -> c_6(zero2#(0(),nil()))
          7: zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
          8: zero2#(0(),nil()) -> c_8()
          9: zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
          10: zero2#(s(y),nil()) -> c_10(zero#(nil()))
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            sub#(s(x),s(y)) -> c_4(sub#(x,y))
            zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
            zero#(nil()) -> c_6(zero2#(0(),nil()))
            zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
            zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
            zero2#(s(y),nil()) -> c_10(zero#(nil()))
        - Weak DPs:
            sub#(0(),0()) -> c_1()
            sub#(0(),s(x)) -> c_2()
            sub#(s(x),0()) -> c_3()
            zero2#(0(),nil()) -> c_8()
        - Weak TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
        - Signature:
            {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
            ,c_7/2,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3}
        by application of
          Pre({3}) = {4,6}.
        Here rules are labelled as follows:
          1: sub#(s(x),s(y)) -> c_4(sub#(x,y))
          2: zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
          3: zero#(nil()) -> c_6(zero2#(0(),nil()))
          4: zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
          5: zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
          6: zero2#(s(y),nil()) -> c_10(zero#(nil()))
          7: sub#(0(),0()) -> c_1()
          8: sub#(0(),s(x)) -> c_2()
          9: sub#(s(x),0()) -> c_3()
          10: zero2#(0(),nil()) -> c_8()
* Step 5: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            sub#(s(x),s(y)) -> c_4(sub#(x,y))
            zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
            zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
            zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
            zero2#(s(y),nil()) -> c_10(zero#(nil()))
        - Weak DPs:
            sub#(0(),0()) -> c_1()
            sub#(0(),s(x)) -> c_2()
            sub#(s(x),0()) -> c_3()
            zero#(nil()) -> c_6(zero2#(0(),nil()))
            zero2#(0(),nil()) -> c_8()
        - Weak TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
        - Signature:
            {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
            ,c_7/2,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {5}
        by application of
          Pre({5}) = {}.
        Here rules are labelled as follows:
          1: sub#(s(x),s(y)) -> c_4(sub#(x,y))
          2: zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
          3: zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
          4: zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
          5: zero2#(s(y),nil()) -> c_10(zero#(nil()))
          6: sub#(0(),0()) -> c_1()
          7: sub#(0(),s(x)) -> c_2()
          8: sub#(s(x),0()) -> c_3()
          9: zero#(nil()) -> c_6(zero2#(0(),nil()))
          10: zero2#(0(),nil()) -> c_8()
* Step 6: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            sub#(s(x),s(y)) -> c_4(sub#(x,y))
            zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
            zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
            zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
        - Weak DPs:
            sub#(0(),0()) -> c_1()
            sub#(0(),s(x)) -> c_2()
            sub#(s(x),0()) -> c_3()
            zero#(nil()) -> c_6(zero2#(0(),nil()))
            zero2#(0(),nil()) -> c_8()
            zero2#(s(y),nil()) -> c_10(zero#(nil()))
        - Weak TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
        - Signature:
            {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
            ,c_7/2,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sub#(s(x),s(y)) -> c_4(sub#(x,y))
             -->_1 sub#(s(x),0()) -> c_3():7
             -->_1 sub#(0(),s(x)) -> c_2():6
             -->_1 sub#(0(),0()) -> c_1():5
             -->_1 sub#(s(x),s(y)) -> c_4(sub#(x,y)):1
          
          2:S:zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
             -->_1 zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs))):4
             -->_1 zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs)):3
             -->_2 sub#(0(),0()) -> c_1():5
             -->_2 sub#(s(x),s(y)) -> c_4(sub#(x,y)):1
          
          3:S:zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
             -->_2 zero#(nil()) -> c_6(zero2#(0(),nil())):8
             -->_1 sub#(0(),0()) -> c_1():5
             -->_2 zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x)):2
             -->_1 sub#(s(x),s(y)) -> c_4(sub#(x,y)):1
          
          4:S:zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
             -->_1 zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x)):2
          
          5:W:sub#(0(),0()) -> c_1()
             
          
          6:W:sub#(0(),s(x)) -> c_2()
             
          
          7:W:sub#(s(x),0()) -> c_3()
             
          
          8:W:zero#(nil()) -> c_6(zero2#(0(),nil()))
             -->_1 zero2#(0(),nil()) -> c_8():9
          
          9:W:zero2#(0(),nil()) -> c_8()
             
          
          10:W:zero2#(s(y),nil()) -> c_10(zero#(nil()))
             -->_1 zero#(nil()) -> c_6(zero2#(0(),nil())):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: zero2#(s(y),nil()) -> c_10(zero#(nil()))
          8: zero#(nil()) -> c_6(zero2#(0(),nil()))
          9: zero2#(0(),nil()) -> c_8()
          5: sub#(0(),0()) -> c_1()
          6: sub#(0(),s(x)) -> c_2()
          7: sub#(s(x),0()) -> c_3()
* Step 7: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            sub#(s(x),s(y)) -> c_4(sub#(x,y))
            zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
            zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
            zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
        - Weak TRS:
            sub(0(),0()) -> 0()
            sub(0(),s(x)) -> 0()
            sub(s(x),0()) -> s(x)
            sub(s(x),s(y)) -> sub(x,y)
        - Signature:
            {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
            ,c_7/2,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(zero2#) = {1},
            uargs(c_4) = {1},
            uargs(c_5) = {1,2},
            uargs(c_7) = {1,2},
            uargs(c_9) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [3]                  
              p(cons) = [0]                  
               p(nil) = [0]                  
                 p(s) = [3]                  
               p(sub) = [3]                  
              p(zero) = [0]                  
             p(zero2) = [0]                  
              p(sub#) = [0]                  
             p(zero#) = [0]                  
            p(zero2#) = [1] x1 + [0]         
               p(c_1) = [0]                  
               p(c_2) = [1]                  
               p(c_3) = [0]                  
               p(c_4) = [1] x1 + [0]         
               p(c_5) = [1] x1 + [1] x2 + [0]
               p(c_6) = [0]                  
               p(c_7) = [1] x1 + [1] x2 + [0]
               p(c_8) = [0]                  
               p(c_9) = [1] x1 + [0]         
              p(c_10) = [0]                  
          
          Following rules are strictly oriented:
           zero2#(0(),cons(x,xs)) = [3]                     
                                  > [0]                     
                                  = c_7(sub#(x,x),zero#(xs))
          
          zero2#(s(y),cons(x,xs)) = [3]                     
                                  > [0]                     
                                  = c_9(zero#(cons(x,xs)))  
          
          
          Following rules are (at-least) weakly oriented:
            sub#(s(x),s(y)) =  [0]                                       
                            >= [0]                                       
                            =  c_4(sub#(x,y))                            
          
          zero#(cons(x,xs)) =  [0]                                       
                            >= [3]                                       
                            =  c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
          
               sub(0(),0()) =  [3]                                       
                            >= [3]                                       
                            =  0()                                       
          
              sub(0(),s(x)) =  [3]                                       
                            >= [3]                                       
                            =  0()                                       
          
              sub(s(x),0()) =  [3]                                       
                            >= [3]                                       
                            =  s(x)                                      
          
             sub(s(x),s(y)) =  [3]                                       
                            >= [3]                                       
                            =  sub(x,y)                                  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          sub#(s(x),s(y)) -> c_4(sub#(x,y))
          zero#(cons(x,xs)) -> c_5(zero2#(sub(x,x),cons(x,xs)),sub#(x,x))
      - Weak DPs:
          zero2#(0(),cons(x,xs)) -> c_7(sub#(x,x),zero#(xs))
          zero2#(s(y),cons(x,xs)) -> c_9(zero#(cons(x,xs)))
      - Weak TRS:
          sub(0(),0()) -> 0()
          sub(0(),s(x)) -> 0()
          sub(s(x),0()) -> s(x)
          sub(s(x),s(y)) -> sub(x,y)
      - Signature:
          {sub/2,zero/1,zero2/2,sub#/2,zero#/1,zero2#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/0,c_4/1,c_5/2,c_6/1
          ,c_7/2,c_8/0,c_9/1,c_10/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {sub#,zero#,zero2#} and constructors {0,cons,nil,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE