MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            carry#2(x2,Nil()) -> Cons(x2,Nil())
            carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            main(x2,x1) -> carry#2(x2,x1)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2,carry#2,cond_carry_w_xs_1,lt#2,main
            ,mult#2} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          addNat#2#(0(),x16) -> c_1()
          addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
          carry#2#(x2,Nil()) -> c_3()
          carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
          cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
          lt#2#(0(),0()) -> c_7()
          lt#2#(0(),S(x16)) -> c_8()
          lt#2#(S(x16),0()) -> c_9()
          lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
          main#(x2,x1) -> c_11(carry#2#(x2,x1))
          mult#2#(0(),x2) -> c_12()
          mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(0(),x16) -> c_1()
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
            carry#2#(x2,Nil()) -> c_3()
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
            lt#2#(0(),0()) -> c_7()
            lt#2#(0(),S(x16)) -> c_8()
            lt#2#(S(x16),0()) -> c_9()
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            main#(x2,x1) -> c_11(carry#2#(x2,x1))
            mult#2#(0(),x2) -> c_12()
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            carry#2(x2,Nil()) -> Cons(x2,Nil())
            carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            main(x2,x1) -> carry#2(x2,x1)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          addNat#2(0(),x16) -> x16
          addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
          lt#2(0(),0()) -> False()
          lt#2(0(),S(x16)) -> True()
          lt#2(S(x16),0()) -> False()
          lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
          mult#2(0(),x2) -> 0()
          mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
          addNat#2#(0(),x16) -> c_1()
          addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
          carry#2#(x2,Nil()) -> c_3()
          carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
          cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
          lt#2#(0(),0()) -> c_7()
          lt#2#(0(),S(x16)) -> c_8()
          lt#2#(S(x16),0()) -> c_9()
          lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
          main#(x2,x1) -> c_11(carry#2#(x2,x1))
          mult#2#(0(),x2) -> c_12()
          mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(0(),x16) -> c_1()
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
            carry#2#(x2,Nil()) -> c_3()
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
            lt#2#(0(),0()) -> c_7()
            lt#2#(0(),S(x16)) -> c_8()
            lt#2#(S(x16),0()) -> c_9()
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            main#(x2,x1) -> c_11(carry#2#(x2,x1))
            mult#2#(0(),x2) -> c_12()
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,6,7,8,9,12}
        by application of
          Pre({1,3,6,7,8,9,12}) = {2,4,5,10,11,13}.
        Here rules are labelled as follows:
          1: addNat#2#(0(),x16) -> c_1()
          2: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
          3: carry#2#(x2,Nil()) -> c_3()
          4: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
          5: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
          6: cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
          7: lt#2#(0(),0()) -> c_7()
          8: lt#2#(0(),S(x16)) -> c_8()
          9: lt#2#(S(x16),0()) -> c_9()
          10: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
          11: main#(x2,x1) -> c_11(carry#2#(x2,x1))
          12: mult#2#(0(),x2) -> c_12()
          13: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            main#(x2,x1) -> c_11(carry#2#(x2,x1))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak DPs:
            addNat#2#(0(),x16) -> c_1()
            carry#2#(x2,Nil()) -> c_3()
            cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
            lt#2#(0(),0()) -> c_7()
            lt#2#(0(),S(x16)) -> c_8()
            lt#2#(S(x16),0()) -> c_9()
            mult#2#(0(),x2) -> c_12()
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
             -->_1 addNat#2#(0(),x16) -> c_1():7
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
          
          2:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):3
             -->_2 lt#2#(S(x16),0()) -> c_9():12
             -->_2 lt#2#(0(),S(x16)) -> c_8():11
             -->_2 lt#2#(0(),0()) -> c_7():10
             -->_1 cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6():9
          
          3:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
             -->_1 carry#2#(x2,Nil()) -> c_3():8
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2
          
          4:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x16),0()) -> c_9():12
             -->_1 lt#2#(0(),S(x16)) -> c_8():11
             -->_1 lt#2#(0(),0()) -> c_7():10
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4
          
          5:S:main#(x2,x1) -> c_11(carry#2#(x2,x1))
             -->_1 carry#2#(x2,Nil()) -> c_3():8
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2
          
          6:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
             -->_2 mult#2#(0(),x2) -> c_12():13
             -->_1 addNat#2#(0(),x16) -> c_1():7
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
          
          7:W:addNat#2#(0(),x16) -> c_1()
             
          
          8:W:carry#2#(x2,Nil()) -> c_3()
             
          
          9:W:cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
             
          
          10:W:lt#2#(0(),0()) -> c_7()
             
          
          11:W:lt#2#(0(),S(x16)) -> c_8()
             
          
          12:W:lt#2#(S(x16),0()) -> c_9()
             
          
          13:W:mult#2#(0(),x2) -> c_12()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6()
          8: carry#2#(x2,Nil()) -> c_3()
          13: mult#2#(0(),x2) -> c_12()
          10: lt#2#(0(),0()) -> c_7()
          11: lt#2#(0(),S(x16)) -> c_8()
          12: lt#2#(S(x16),0()) -> c_9()
          7: addNat#2#(0(),x16) -> c_1()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            main#(x2,x1) -> c_11(carry#2#(x2,x1))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
           -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
        
        2:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
           -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4
           -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)):3
        
        3:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
           -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
           -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2
        
        4:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
           -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4
        
        5:S:main#(x2,x1) -> c_11(carry#2#(x2,x1))
           -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2
        
        6:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
           -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
           -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):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).
        
        [(5,main#(x2,x1) -> c_11(carry#2#(x2,x1)))]
* Step 6: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
          - Weak DPs:
              carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
              cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
              lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
              mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
          - Weak TRS:
              addNat#2(0(),x16) -> x16
              addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
              lt#2(0(),0()) -> False()
              lt#2(0(),S(x16)) -> True()
              lt#2(S(x16),0()) -> False()
              lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
              mult#2(0(),x2) -> 0()
              mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
          - Signature:
              {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2
              ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0
              ,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
              ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
              cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
              lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
              mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
          - Weak DPs:
              addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
          - Weak TRS:
              addNat#2(0(),x16) -> x16
              addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
              lt#2(0(),0()) -> False()
              lt#2(0(),S(x16)) -> True()
              lt#2(S(x16),0()) -> False()
              lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
              mult#2(0(),x2) -> 0()
              mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
          - Signature:
              {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2
              ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0
              ,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
              ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
** Step 6.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
          
          2:W:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):3
          
          3:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
          
          4:W:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4
          
          6:W:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
** Step 6.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
          
          2:W:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):3
          
          3:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
          
          6:W:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
** Step 6.a:3: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        and a lower component
          addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
          mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        Further, following extension rules are added to the lower component.
          carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
*** Step 6.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
          
        Consider the set of all dependency pairs
          1: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
          2: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
        Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_4) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#}
        TcT has computed the following interpretation:
                           p(0) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                        p(Cons) = [0 0 1]      [0 0 1]      [0]             
                                  [0 0 0] x1 + [0 0 1] x2 + [0]             
                                  [0 0 1]      [0 0 1]      [1]             
                       p(False) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(Nil) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                           p(S) = [0 1 0]      [0]                          
                                  [0 0 1] x1 + [1]                          
                                  [0 0 0]      [1]                          
                        p(True) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                    p(addNat#2) = [0 0 0]      [1 0 0]      [1]             
                                  [1 1 1] x1 + [0 0 0] x2 + [1]             
                                  [0 1 1]      [0 1 1]      [1]             
                     p(carry#2) = [0]                                       
                                  [0]                                       
                                  [0]                                       
           p(cond_carry_w_xs_1) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                        p(lt#2) = [0 0 0]      [0]                          
                                  [0 0 0] x1 + [0]                          
                                  [1 0 0]      [0]                          
                        p(main) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                      p(mult#2) = [0 0 0]      [0 1 0]      [1]             
                                  [0 0 0] x1 + [0 1 0] x2 + [0]             
                                  [0 1 0]      [0 0 0]      [0]             
                   p(addNat#2#) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                    p(carry#2#) = [0 0 1]      [0]                          
                                  [1 0 1] x2 + [0]                          
                                  [0 1 1]      [0]                          
          p(cond_carry_w_xs_1#) = [0 0 0]      [0 0 1]      [0 0 1]      [1]
                                  [0 0 1] x2 + [0 0 0] x3 + [1 0 1] x4 + [1]
                                  [0 0 1]      [0 0 0]      [1 1 1]      [1]
                       p(lt#2#) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                       p(main#) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                     p(mult#2#) = [0 0 1]      [0]                          
                                  [0 0 1] x1 + [0]                          
                                  [0 0 1]      [1]                          
                         p(c_1) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(c_2) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(c_3) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(c_4) = [1 0 0]      [0]                          
                                  [0 0 0] x1 + [1]                          
                                  [1 0 0]      [0]                          
                         p(c_5) = [1 0 0]      [0 0 0]      [0]             
                                  [0 1 0] x1 + [0 0 0] x2 + [1]             
                                  [0 0 1]      [1 0 0]      [0]             
                         p(c_6) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(c_7) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(c_8) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                         p(c_9) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                        p(c_10) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                        p(c_11) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                        p(c_12) = [0]                                       
                                  [0]                                       
                                  [0]                                       
                        p(c_13) = [0]                                       
                                  [0]                                       
                                  [0]                                       
        
        Following rules are strictly oriented:
        cond_carry_w_xs_1#(False(),x3,x2,x1) = [0 0 1]      [0 0 1]      [0 0 0]      [1]                  
                                               [1 0 1] x1 + [0 0 0] x2 + [0 0 1] x3 + [1]                  
                                               [1 1 1]      [0 0 0]      [0 0 1]      [1]                  
                                             > [0 0 1]      [0]                                            
                                               [1 0 1] x1 + [1]                                            
                                               [0 1 1]      [1]                                            
                                             = c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        
        
        Following rules are (at-least) weakly oriented:
        carry#2#(x6,Cons(x4,x2)) =  [0 0 1]      [0 0 1]      [1]                
                                    [0 0 2] x2 + [0 0 2] x4 + [1]                
                                    [0 0 2]      [0 0 1]      [1]                
                                 >= [0 0 1]      [0 0 1]      [1]                
                                    [0 0 0] x2 + [0 0 0] x4 + [1]                
                                    [0 0 1]      [0 0 1]      [1]                
                                 =  c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
        
**** Step 6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):2
          
          2:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2))
          2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
**** Step 6.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 6.a:3.b:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
          mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        and a lower component
          addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
        Further, following extension rules are added to the lower component.
          carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
          mult#2#(S(x4),x2) -> addNat#2#(mult#2(x4,x2),x2)
          mult#2#(S(x4),x2) -> mult#2#(x4,x2)
**** Step 6.a:3.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
          
        The strictly oriented rules are moved into the weak component.
***** Step 6.a:3.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_13) = {1,2}
        
        Following symbols are considered usable:
          {lt#2,addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#}
        TcT has computed the following interpretation:
                           p(0) = [10]                                   
                        p(Cons) = [1] x1 + [1] x2 + [2]                  
                       p(False) = [2]                                    
                         p(Nil) = [1]                                    
                           p(S) = [1] x1 + [8]                           
                        p(True) = [1]                                    
                    p(addNat#2) = [1] x2 + [14]                          
                     p(carry#2) = [1] x1 + [1]                           
           p(cond_carry_w_xs_1) = [8] x1 + [1] x2 + [1] x3 + [1] x4 + [2]
                        p(lt#2) = [1] x2 + [1]                           
                        p(main) = [1]                                    
                      p(mult#2) = [6]                                    
                   p(addNat#2#) = [2]                                    
                    p(carry#2#) = [8] x2 + [4]                           
          p(cond_carry_w_xs_1#) = [8] x1 + [8] x4 + [12]                 
                       p(lt#2#) = [1] x1 + [0]                           
                       p(main#) = [1]                                    
                     p(mult#2#) = [1] x1 + [2]                           
                         p(c_1) = [1]                                    
                         p(c_2) = [0]                                    
                         p(c_3) = [1]                                    
                         p(c_4) = [4] x1 + [4]                           
                         p(c_5) = [1] x2 + [1]                           
                         p(c_6) = [2]                                    
                         p(c_7) = [0]                                    
                         p(c_8) = [1]                                    
                         p(c_9) = [1]                                    
                        p(c_10) = [1] x1 + [0]                           
                        p(c_11) = [0]                                    
                        p(c_12) = [0]                                    
                        p(c_13) = [1] x1 + [1] x2 + [4]                  
        
        Following rules are strictly oriented:
        mult#2#(S(x4),x2) = [1] x4 + [10]                                   
                          > [1] x4 + [8]                                    
                          = c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        
        
        Following rules are (at-least) weakly oriented:
                    carry#2#(x6,Cons(x4,x2)) =  [8] x2 + [8] x4 + [20]                  
                                             >= [8] x2 + [8] x4 + [20]                  
                                             =  cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
        
        cond_carry_w_xs_1#(False(),x3,x2,x1) =  [8] x1 + [28]                           
                                             >= [8] x1 + [4]                            
                                             =  carry#2#(mult#2(S(S(0())),x3),x1)       
        
        cond_carry_w_xs_1#(False(),x3,x2,x1) =  [8] x1 + [28]                           
                                             >= [28]                                    
                                             =  mult#2#(S(S(0())),x3)                   
        
                               lt#2(0(),0()) =  [11]                                    
                                             >= [2]                                     
                                             =  False()                                 
        
                            lt#2(0(),S(x16)) =  [1] x16 + [9]                           
                                             >= [1]                                     
                                             =  True()                                  
        
                            lt#2(S(x16),0()) =  [11]                                    
                                             >= [2]                                     
                                             =  False()                                 
        
                           lt#2(S(x4),S(x2)) =  [1] x2 + [9]                            
                                             >= [1] x2 + [1]                            
                                             =  lt#2(x4,x2)                             
        
***** Step 6.a:3.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 6.a:3.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3):3
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1):2
          
          2:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
             -->_1 carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2):1
          
          3:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
             -->_1 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
          
          4:W:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
          2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
          3: cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
          4: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
***** Step 6.a:3.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 6.a:3.b:1.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
      - Weak DPs:
          carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1)
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3)
          mult#2#(S(x4),x2) -> addNat#2#(mult#2(x4,x2),x2)
          mult#2#(S(x4),x2) -> mult#2#(x4,x2)
      - Weak TRS:
          addNat#2(0(),x16) -> x16
          addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
          lt#2(0(),0()) -> False()
          lt#2(0(),S(x16)) -> True()
          lt#2(S(x16),0()) -> False()
          lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
          mult#2(0(),x2) -> 0()
          mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
      - Signature:
          {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
          ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0
          ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
          ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak DPs:
            addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):2
          
          2:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1
          
          3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
          
          4:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):5
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
          
          5:W:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
             -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):2
          
          2:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1
          
          3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
          
          4:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
             -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          4: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
            mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_4) = {1,2},
          uargs(c_5) = {1,2},
          uargs(c_10) = {1},
          uargs(c_13) = {1}
        
        Following symbols are considered usable:
          {lt#2,addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#}
        TcT has computed the following interpretation:
                           p(0) = [0]                    
                        p(Cons) = [1] x1 + [1] x2 + [2]  
                       p(False) = [2]                    
                         p(Nil) = [1]                    
                           p(S) = [1] x1 + [4]           
                        p(True) = [6]                    
                    p(addNat#2) = [4] x2 + [4]           
                     p(carry#2) = [1] x2 + [0]           
           p(cond_carry_w_xs_1) = [4] x2 + [2] x4 + [0]  
                        p(lt#2) = [1] x2 + [2]           
                        p(main) = [1] x2 + [0]           
                      p(mult#2) = [0]                    
                   p(addNat#2#) = [0]                    
                    p(carry#2#) = [10] x2 + [8]          
          p(cond_carry_w_xs_1#) = [10] x1 + [10] x4 + [8]
                       p(lt#2#) = [0]                    
                       p(main#) = [1] x1 + [1] x2 + [0]  
                     p(mult#2#) = [1] x1 + [2]           
                         p(c_1) = [2]                    
                         p(c_2) = [1] x1 + [0]           
                         p(c_3) = [2]                    
                         p(c_4) = [1] x1 + [8] x2 + [0]  
                         p(c_5) = [1] x1 + [2] x2 + [0]  
                         p(c_6) = [0]                    
                         p(c_7) = [8]                    
                         p(c_8) = [4]                    
                         p(c_9) = [0]                    
                        p(c_10) = [2] x1 + [0]           
                        p(c_11) = [1] x1 + [0]           
                        p(c_12) = [0]                    
                        p(c_13) = [1] x1 + [0]           
        
        Following rules are strictly oriented:
        mult#2#(S(x4),x2) = [1] x4 + [6]        
                          > [1] x4 + [2]        
                          = c_13(mult#2#(x4,x2))
        
        
        Following rules are (at-least) weakly oriented:
                    carry#2#(x6,Cons(x4,x2)) =  [10] x2 + [10] x4 + [28]                                    
                                             >= [10] x2 + [10] x4 + [28]                                    
                                             =  c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))  
        
        cond_carry_w_xs_1#(False(),x3,x2,x1) =  [10] x1 + [28]                                              
                                             >= [10] x1 + [28]                                              
                                             =  c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
        
                          lt#2#(S(x4),S(x2)) =  [0]                                                         
                                             >= [0]                                                         
                                             =  c_10(lt#2#(x4,x2))                                          
        
                               lt#2(0(),0()) =  [2]                                                         
                                             >= [2]                                                         
                                             =  False()                                                     
        
                            lt#2(0(),S(x16)) =  [1] x16 + [6]                                               
                                             >= [6]                                                         
                                             =  True()                                                      
        
                            lt#2(S(x16),0()) =  [2]                                                         
                                             >= [2]                                                         
                                             =  False()                                                     
        
                           lt#2(S(x4),S(x2)) =  [1] x2 + [6]                                                
                                             >= [1] x2 + [2]                                                
                                             =  lt#2(x4,x2)                                                 
        
*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak DPs:
            mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak DPs:
            mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):2
          
          2:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_2 mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)):4
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1
          
          3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
          
          4:W:mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
             -->_1 mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2))
*** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)
                                                              ,mult#2#(S(S(0())),x3)):2
          
          2:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3))
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1
          
          3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
*** Step 6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
          3: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
          
        Consider the set of all dependency pairs
          1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
          2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
          3: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1,3}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_4) = {1,2},
          uargs(c_5) = {1},
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#}
        TcT has computed the following interpretation:
                           p(0) = [8]                  
                        p(Cons) = [1] x1 + [1] x2 + [2]
                       p(False) = [0]                  
                         p(Nil) = [2]                  
                           p(S) = [1] x1 + [1]         
                        p(True) = [1]                  
                    p(addNat#2) = [2] x1 + [8] x2 + [8]
                     p(carry#2) = [0]                  
           p(cond_carry_w_xs_1) = [2] x1 + [1] x3 + [1]
                        p(lt#2) = [0]                  
                        p(main) = [2] x1 + [8] x2 + [1]
                      p(mult#2) = [3]                  
                   p(addNat#2#) = [1] x1 + [1]         
                    p(carry#2#) = [8] x2 + [0]         
          p(cond_carry_w_xs_1#) = [8] x4 + [3]         
                       p(lt#2#) = [4] x2 + [2]         
                       p(main#) = [1] x1 + [1]         
                     p(mult#2#) = [2] x1 + [1]         
                         p(c_1) = [1]                  
                         p(c_2) = [0]                  
                         p(c_3) = [1]                  
                         p(c_4) = [1] x1 + [2] x2 + [1]
                         p(c_5) = [1] x1 + [3]         
                         p(c_6) = [2]                  
                         p(c_7) = [0]                  
                         p(c_8) = [0]                  
                         p(c_9) = [0]                  
                        p(c_10) = [1] x1 + [2]         
                        p(c_11) = [1] x1 + [0]         
                        p(c_12) = [1]                  
                        p(c_13) = [2] x1 + [0]         
        
        Following rules are strictly oriented:
        carry#2#(x6,Cons(x4,x2)) = [8] x2 + [8] x4 + [16]                                    
                                 > [8] x2 + [8] x4 + [8]                                     
                                 = c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
        
              lt#2#(S(x4),S(x2)) = [4] x2 + [6]                                              
                                 > [4] x2 + [4]                                              
                                 = c_10(lt#2#(x4,x2))                                        
        
        
        Following rules are (at-least) weakly oriented:
        cond_carry_w_xs_1#(False(),x3,x2,x1) =  [8] x1 + [3]                          
                                             >= [8] x1 + [3]                          
                                             =  c_5(carry#2#(mult#2(S(S(0())),x3),x1))
        
**** Step 6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
            cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
            lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
             -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
             -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)):2
          
          2:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
             -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1
          
          3:W:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
             -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4))
          2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1))
          3: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2))
**** Step 6.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            addNat#2(0(),x16) -> x16
            addNat#2(S(x4),x2) -> S(addNat#2(x4,x2))
            lt#2(0(),0()) -> False()
            lt#2(0(),S(x16)) -> True()
            lt#2(S(x16),0()) -> False()
            lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
            mult#2(0(),x2) -> 0()
            mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2)
        - Signature:
            {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,cond_carry_w_xs_1#/4
            ,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0
            ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#
            ,mult#2#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE