WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            goal(xs,ys) -> mul0(xs,ys)
            isZero(C(x,y)) -> False()
            isZero(Z()) -> True()
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
            second(C(x,y)) -> y
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1} / {C/2,False/0,S/0,True/0,Z/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0,goal,isZero,mul0,second} and constructors {C,False,S
            ,True,Z}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
          add0#(Z(),y) -> c_2()
          goal#(xs,ys) -> c_3(mul0#(xs,ys))
          isZero#(C(x,y)) -> c_4()
          isZero#(Z()) -> c_5()
          mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
          mul0#(Z(),y) -> c_7()
          second#(C(x,y)) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            add0#(Z(),y) -> c_2()
            goal#(xs,ys) -> c_3(mul0#(xs,ys))
            isZero#(C(x,y)) -> c_4()
            isZero#(Z()) -> c_5()
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
            mul0#(Z(),y) -> c_7()
            second#(C(x,y)) -> c_8()
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            goal(xs,ys) -> mul0(xs,ys)
            isZero(C(x,y)) -> False()
            isZero(Z()) -> True()
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
            second(C(x,y)) -> y
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add0(C(x,y),y') -> add0(y,C(S(),y'))
          add0(Z(),y) -> y
          mul0(C(x,y),y') -> add0(mul0(y,y'),y')
          mul0(Z(),y) -> Z()
          add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
          add0#(Z(),y) -> c_2()
          goal#(xs,ys) -> c_3(mul0#(xs,ys))
          isZero#(C(x,y)) -> c_4()
          isZero#(Z()) -> c_5()
          mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
          mul0#(Z(),y) -> c_7()
          second#(C(x,y)) -> c_8()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            add0#(Z(),y) -> c_2()
            goal#(xs,ys) -> c_3(mul0#(xs,ys))
            isZero#(C(x,y)) -> c_4()
            isZero#(Z()) -> c_5()
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
            mul0#(Z(),y) -> c_7()
            second#(C(x,y)) -> c_8()
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,5,7,8}
        by application of
          Pre({2,4,5,7,8}) = {1,3,6}.
        Here rules are labelled as follows:
          1: add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
          2: add0#(Z(),y) -> c_2()
          3: goal#(xs,ys) -> c_3(mul0#(xs,ys))
          4: isZero#(C(x,y)) -> c_4()
          5: isZero#(Z()) -> c_5()
          6: mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
          7: mul0#(Z(),y) -> c_7()
          8: second#(C(x,y)) -> c_8()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            goal#(xs,ys) -> c_3(mul0#(xs,ys))
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak DPs:
            add0#(Z(),y) -> c_2()
            isZero#(C(x,y)) -> c_4()
            isZero#(Z()) -> c_5()
            mul0#(Z(),y) -> c_7()
            second#(C(x,y)) -> c_8()
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
             -->_1 add0#(Z(),y) -> c_2():4
             -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):1
          
          2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys))
             -->_1 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):3
             -->_1 mul0#(Z(),y) -> c_7():7
          
          3:S:mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
             -->_2 mul0#(Z(),y) -> c_7():7
             -->_1 add0#(Z(),y) -> c_2():4
             -->_2 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):3
             -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):1
          
          4:W:add0#(Z(),y) -> c_2()
             
          
          5:W:isZero#(C(x,y)) -> c_4()
             
          
          6:W:isZero#(Z()) -> c_5()
             
          
          7:W:mul0#(Z(),y) -> c_7()
             
          
          8:W:second#(C(x,y)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: second#(C(x,y)) -> c_8()
          6: isZero#(Z()) -> c_5()
          5: isZero#(C(x,y)) -> c_4()
          7: mul0#(Z(),y) -> c_7()
          4: add0#(Z(),y) -> c_2()
* Step 5: RemoveHeads WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            goal#(xs,ys) -> c_3(mul0#(xs,ys))
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
           -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):1
        
        2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys))
           -->_1 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):3
        
        3:S:mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
           -->_2 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):3
           -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):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).
        
        [(2,goal#(xs,ys) -> c_3(mul0#(xs,ys)))]
* Step 6: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + 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:
              add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
          - Weak DPs:
              mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
          - Weak TRS:
              add0(C(x,y),y') -> add0(y,C(S(),y'))
              add0(Z(),y) -> y
              mul0(C(x,y),y') -> add0(mul0(y,y'),y')
              mul0(Z(),y) -> Z()
          - Signature:
              {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
              ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
              ,False,S,True,Z}
        
        Problem (S)
          - Strict DPs:
              mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
          - Weak DPs:
              add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
          - Weak TRS:
              add0(C(x,y),y') -> add0(y,C(S(),y'))
              add0(Z(),y) -> y
              mul0(C(x,y),y') -> add0(mul0(y,y'),y')
              mul0(Z(),y) -> Z()
          - Signature:
              {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
              ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
              ,False,S,True,Z}
** Step 6.a:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
        - Weak DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + 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
          mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        and a lower component
          add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
        Further, following extension rules are added to the lower component.
          mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
          mul0#(C(x,y),y') -> mul0#(y,y')
*** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + 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: mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + 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_6) = {1,2}
        
        Following symbols are considered usable:
          {add0#,goal#,isZero#,mul0#,second#}
        TcT has computed the following interpretation:
                p(C) = [1] x1 + [1] x2 + [1]
            p(False) = [0]                  
                p(S) = [4]                  
             p(True) = [1]                  
                p(Z) = [0]                  
             p(add0) = [4] x1 + [2] x2 + [2]
             p(goal) = [1] x1 + [2] x2 + [0]
           p(isZero) = [1] x1 + [2]         
             p(mul0) = [2] x1 + [1] x2 + [1]
           p(second) = [2] x1 + [1]         
            p(add0#) = [2]                  
            p(goal#) = [2]                  
          p(isZero#) = [4] x1 + [2]         
            p(mul0#) = [8] x1 + [4] x2 + [0]
          p(second#) = [1]                  
              p(c_1) = [1]                  
              p(c_2) = [1]                  
              p(c_3) = [1] x1 + [1]         
              p(c_4) = [1]                  
              p(c_5) = [1]                  
              p(c_6) = [1] x1 + [1] x2 + [0]
              p(c_7) = [1]                  
              p(c_8) = [2]                  
        
        Following rules are strictly oriented:
        mul0#(C(x,y),y') = [8] x + [8] y + [4] y' + [8]         
                         > [8] y + [4] y' + [2]                 
                         = c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
             -->_2 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
**** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
        - Weak DPs:
            mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
            mul0#(C(x,y),y') -> mul0#(y,y')
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
        - Weak DPs:
            mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
            mul0#(C(x,y),y') -> mul0#(y,y')
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_1) = {1}
        
        Following symbols are considered usable:
          {add0,mul0,add0#,goal#,isZero#,mul0#,second#}
        TcT has computed the following interpretation:
                p(C) = 1 + x2                            
            p(False) = 0                                 
                p(S) = 0                                 
             p(True) = 0                                 
                p(Z) = 0                                 
             p(add0) = x1 + x2                           
             p(goal) = 0                                 
           p(isZero) = 4*x1^2                            
             p(mul0) = x1 + x1*x2                        
           p(second) = 4                                 
            p(add0#) = 1 + x1                            
            p(goal#) = 1 + 2*x1^2 + x2 + 2*x2^2          
          p(isZero#) = 4                                 
            p(mul0#) = 5 + 3*x1 + 4*x1*x2 + 5*x2 + 6*x2^2
          p(second#) = 4 + x1                            
              p(c_1) = x1                                
              p(c_2) = 1                                 
              p(c_3) = 4 + x1                            
              p(c_4) = 1                                 
              p(c_5) = 0                                 
              p(c_6) = 0                                 
              p(c_7) = 1                                 
              p(c_8) = 0                                 
        
        Following rules are strictly oriented:
        add0#(C(x,y),y') = 2 + y                  
                         > 1 + y                  
                         = c_1(add0#(y,C(S(),y')))
        
        
        Following rules are (at-least) weakly oriented:
        mul0#(C(x,y),y') =  8 + 3*y + 4*y*y' + 9*y' + 6*y'^2
                         >= 1 + y + y*y'                    
                         =  add0#(mul0(y,y'),y')            
        
        mul0#(C(x,y),y') =  8 + 3*y + 4*y*y' + 9*y' + 6*y'^2
                         >= 5 + 3*y + 4*y*y' + 5*y' + 6*y'^2
                         =  mul0#(y,y')                     
        
         add0(C(x,y),y') =  1 + y + y'                      
                         >= 1 + y + y'                      
                         =  add0(y,C(S(),y'))               
        
             add0(Z(),y) =  y                               
                         >= y                               
                         =  y                               
        
         mul0(C(x,y),y') =  1 + y + y*y' + y'               
                         >= y + y*y' + y'                   
                         =  add0(mul0(y,y'),y')             
        
             mul0(Z(),y) =  0                               
                         >= 0                               
                         =  Z()                             
        
**** Step 6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
            mul0#(C(x,y),y') -> mul0#(y,y')
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
            mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
            mul0#(C(x,y),y') -> mul0#(y,y')
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
             -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):1
          
          2:W:mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
             -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):1
          
          3:W:mul0#(C(x,y),y') -> mul0#(y,y')
             -->_1 mul0#(C(x,y),y') -> mul0#(y,y'):3
             -->_1 mul0#(C(x,y),y') -> add0#(mul0(y,y'),y'):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: mul0#(C(x,y),y') -> mul0#(y,y')
          2: mul0#(C(x,y),y') -> add0#(mul0(y,y'),y')
          1: add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
**** Step 6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak DPs:
            add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
             -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):2
             -->_2 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):1
          
          2:W:add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
             -->_1 add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y'))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: add0#(C(x,y),y') -> c_1(add0#(y,C(S(),y')))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y'))
             -->_2 mul0#(C(x,y),y') -> c_6(add0#(mul0(y,y'),y'),mul0#(y,y')):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
** Step 6.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
        - Weak TRS:
            add0(C(x,y),y') -> add0(y,C(S(),y'))
            add0(Z(),y) -> y
            mul0(C(x,y),y') -> add0(mul0(y,y'),y')
            mul0(Z(),y) -> Z()
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
** Step 6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + 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: mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + 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_6) = {1}
        
        Following symbols are considered usable:
          {add0#,goal#,isZero#,mul0#,second#}
        TcT has computed the following interpretation:
                p(C) = [1] x1 + [1] x2 + [2]
            p(False) = [1]                  
                p(S) = [0]                  
             p(True) = [1]                  
                p(Z) = [2]                  
             p(add0) = [4] x1 + [1] x2 + [2]
             p(goal) = [1]                  
           p(isZero) = [1] x1 + [2]         
             p(mul0) = [4] x2 + [0]         
           p(second) = [1] x1 + [1]         
            p(add0#) = [1] x1 + [1]         
            p(goal#) = [1] x1 + [2] x2 + [1]
          p(isZero#) = [8]                  
            p(mul0#) = [8] x1 + [1] x2 + [0]
          p(second#) = [1] x1 + [1]         
              p(c_1) = [8]                  
              p(c_2) = [0]                  
              p(c_3) = [2]                  
              p(c_4) = [8]                  
              p(c_5) = [0]                  
              p(c_6) = [1] x1 + [11]        
              p(c_7) = [0]                  
              p(c_8) = [2]                  
        
        Following rules are strictly oriented:
        mul0#(C(x,y),y') = [8] x + [8] y + [1] y' + [16]
                         > [8] y + [1] y' + [11]        
                         = c_6(mul0#(y,y'))             
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 6.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
             -->_1 mul0#(C(x,y),y') -> c_6(mul0#(y,y')):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: mul0#(C(x,y),y') -> c_6(mul0#(y,y'))
*** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {add0/2,goal/2,isZero/1,mul0/2,second/1,add0#/2,goal#/2,isZero#/1,mul0#/2,second#/1} / {C/2,False/0,S/0
            ,True/0,Z/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add0#,goal#,isZero#,mul0#,second#} and constructors {C
            ,False,S,True,Z}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))