MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            adx(cons(X,L)) -> incr(cons(X,adx(L)))
            adx(nil()) -> nil()
            head(cons(X,L)) -> X
            incr(cons(X,L)) -> cons(s(X),incr(L))
            incr(nil()) -> nil()
            nats() -> adx(zeros())
            tail(cons(X,L)) -> L
            zeros() -> cons(0(),zeros())
        - Signature:
            {adx/1,head/1,incr/1,nats/0,tail/1,zeros/0} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {adx,head,incr,nats,tail,zeros} and constructors {0,cons
            ,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
          adx#(nil()) -> c_2()
          head#(cons(X,L)) -> c_3()
          incr#(cons(X,L)) -> c_4(incr#(L))
          incr#(nil()) -> c_5()
          nats#() -> c_6(adx#(zeros()),zeros#())
          tail#(cons(X,L)) -> c_7()
          zeros#() -> c_8(zeros#())
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
            adx#(nil()) -> c_2()
            head#(cons(X,L)) -> c_3()
            incr#(cons(X,L)) -> c_4(incr#(L))
            incr#(nil()) -> c_5()
            nats#() -> c_6(adx#(zeros()),zeros#())
            tail#(cons(X,L)) -> c_7()
            zeros#() -> c_8(zeros#())
        - Weak TRS:
            adx(cons(X,L)) -> incr(cons(X,adx(L)))
            adx(nil()) -> nil()
            head(cons(X,L)) -> X
            incr(cons(X,L)) -> cons(s(X),incr(L))
            incr(nil()) -> nil()
            nats() -> adx(zeros())
            tail(cons(X,L)) -> L
            zeros() -> cons(0(),zeros())
        - Signature:
            {adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,adx#/1,head#/1,incr#/1,nats#/0,tail#/1,zeros#/0} / {0/0,cons/2
            ,nil/0,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {adx#,head#,incr#,nats#,tail#,zeros#} and constructors {0
            ,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          adx(cons(X,L)) -> incr(cons(X,adx(L)))
          adx(nil()) -> nil()
          incr(cons(X,L)) -> cons(s(X),incr(L))
          incr(nil()) -> nil()
          zeros() -> cons(0(),zeros())
          adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
          adx#(nil()) -> c_2()
          head#(cons(X,L)) -> c_3()
          incr#(cons(X,L)) -> c_4(incr#(L))
          incr#(nil()) -> c_5()
          nats#() -> c_6(adx#(zeros()),zeros#())
          tail#(cons(X,L)) -> c_7()
          zeros#() -> c_8(zeros#())
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
            adx#(nil()) -> c_2()
            head#(cons(X,L)) -> c_3()
            incr#(cons(X,L)) -> c_4(incr#(L))
            incr#(nil()) -> c_5()
            nats#() -> c_6(adx#(zeros()),zeros#())
            tail#(cons(X,L)) -> c_7()
            zeros#() -> c_8(zeros#())
        - Weak TRS:
            adx(cons(X,L)) -> incr(cons(X,adx(L)))
            adx(nil()) -> nil()
            incr(cons(X,L)) -> cons(s(X),incr(L))
            incr(nil()) -> nil()
            zeros() -> cons(0(),zeros())
        - Signature:
            {adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,adx#/1,head#/1,incr#/1,nats#/0,tail#/1,zeros#/0} / {0/0,cons/2
            ,nil/0,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {adx#,head#,incr#,nats#,tail#,zeros#} and constructors {0
            ,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,5,7}
        by application of
          Pre({2,3,5,7}) = {1,4,6}.
        Here rules are labelled as follows:
          1: adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
          2: adx#(nil()) -> c_2()
          3: head#(cons(X,L)) -> c_3()
          4: incr#(cons(X,L)) -> c_4(incr#(L))
          5: incr#(nil()) -> c_5()
          6: nats#() -> c_6(adx#(zeros()),zeros#())
          7: tail#(cons(X,L)) -> c_7()
          8: zeros#() -> c_8(zeros#())
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
            incr#(cons(X,L)) -> c_4(incr#(L))
            nats#() -> c_6(adx#(zeros()),zeros#())
            zeros#() -> c_8(zeros#())
        - Weak DPs:
            adx#(nil()) -> c_2()
            head#(cons(X,L)) -> c_3()
            incr#(nil()) -> c_5()
            tail#(cons(X,L)) -> c_7()
        - Weak TRS:
            adx(cons(X,L)) -> incr(cons(X,adx(L)))
            adx(nil()) -> nil()
            incr(cons(X,L)) -> cons(s(X),incr(L))
            incr(nil()) -> nil()
            zeros() -> cons(0(),zeros())
        - Signature:
            {adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,adx#/1,head#/1,incr#/1,nats#/0,tail#/1,zeros#/0} / {0/0,cons/2
            ,nil/0,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {adx#,head#,incr#,nats#,tail#,zeros#} and constructors {0
            ,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
             -->_1 incr#(cons(X,L)) -> c_4(incr#(L)):2
             -->_2 adx#(nil()) -> c_2():5
             -->_2 adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L)):1
          
          2:S:incr#(cons(X,L)) -> c_4(incr#(L))
             -->_1 incr#(nil()) -> c_5():7
             -->_1 incr#(cons(X,L)) -> c_4(incr#(L)):2
          
          3:S:nats#() -> c_6(adx#(zeros()),zeros#())
             -->_2 zeros#() -> c_8(zeros#()):4
             -->_1 adx#(nil()) -> c_2():5
             -->_1 adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L)):1
          
          4:S:zeros#() -> c_8(zeros#())
             -->_1 zeros#() -> c_8(zeros#()):4
          
          5:W:adx#(nil()) -> c_2()
             
          
          6:W:head#(cons(X,L)) -> c_3()
             
          
          7:W:incr#(nil()) -> c_5()
             
          
          8:W:tail#(cons(X,L)) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: tail#(cons(X,L)) -> c_7()
          6: head#(cons(X,L)) -> c_3()
          5: adx#(nil()) -> c_2()
          7: incr#(nil()) -> c_5()
* Step 5: NaturalMI MAYBE
    + Considered Problem:
        - Strict DPs:
            adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
            incr#(cons(X,L)) -> c_4(incr#(L))
            nats#() -> c_6(adx#(zeros()),zeros#())
            zeros#() -> c_8(zeros#())
        - Weak TRS:
            adx(cons(X,L)) -> incr(cons(X,adx(L)))
            adx(nil()) -> nil()
            incr(cons(X,L)) -> cons(s(X),incr(L))
            incr(nil()) -> nil()
            zeros() -> cons(0(),zeros())
        - Signature:
            {adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,adx#/1,head#/1,incr#/1,nats#/0,tail#/1,zeros#/0} / {0/0,cons/2
            ,nil/0,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {adx#,head#,incr#,nats#,tail#,zeros#} and constructors {0
            ,cons,nil,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        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(c_1) = {1,2},
          uargs(c_4) = {1},
          uargs(c_6) = {1,2},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {zeros,adx#,head#,incr#,nats#,tail#,zeros#}
        TcT has computed the following interpretation:
               p(0) = [1]                  
             p(adx) = [3]                  
            p(cons) = [10]                 
            p(head) = [1] x1 + [1]         
            p(incr) = [1] x1 + [8]         
            p(nats) = [0]                  
             p(nil) = [8]                  
               p(s) = [1]                  
            p(tail) = [0]                  
           p(zeros) = [13]                 
            p(adx#) = [0]                  
           p(head#) = [1]                  
           p(incr#) = [0]                  
           p(nats#) = [9]                  
           p(tail#) = [4] x1 + [1]         
          p(zeros#) = [0]                  
             p(c_1) = [4] x1 + [8] x2 + [0]
             p(c_2) = [0]                  
             p(c_3) = [1]                  
             p(c_4) = [4] x1 + [0]         
             p(c_5) = [0]                  
             p(c_6) = [8] x1 + [2] x2 + [2]
             p(c_7) = [0]                  
             p(c_8) = [8] x1 + [0]         
        
        Following rules are strictly oriented:
        nats#() = [9]                        
                > [2]                        
                = c_6(adx#(zeros()),zeros#())
        
        
        Following rules are (at-least) weakly oriented:
         adx#(cons(X,L)) =  [0]                               
                         >= [0]                               
                         =  c_1(incr#(cons(X,adx(L))),adx#(L))
        
        incr#(cons(X,L)) =  [0]                               
                         >= [0]                               
                         =  c_4(incr#(L))                     
        
                zeros#() =  [0]                               
                         >= [0]                               
                         =  c_8(zeros#())                     
        
                 zeros() =  [13]                              
                         >= [10]                              
                         =  cons(0(),zeros())                 
        
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          adx#(cons(X,L)) -> c_1(incr#(cons(X,adx(L))),adx#(L))
          incr#(cons(X,L)) -> c_4(incr#(L))
          zeros#() -> c_8(zeros#())
      - Weak DPs:
          nats#() -> c_6(adx#(zeros()),zeros#())
      - Weak TRS:
          adx(cons(X,L)) -> incr(cons(X,adx(L)))
          adx(nil()) -> nil()
          incr(cons(X,L)) -> cons(s(X),incr(L))
          incr(nil()) -> nil()
          zeros() -> cons(0(),zeros())
      - Signature:
          {adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,adx#/1,head#/1,incr#/1,nats#/0,tail#/1,zeros#/0} / {0/0,cons/2
          ,nil/0,s/1,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {adx#,head#,incr#,nats#,tail#,zeros#} and constructors {0
          ,cons,nil,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE