WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(C(x1,x2),y,z) -> C(a(x1,y,z),a(x2,y,y))
            a(Z(),y,z) -> Z()
            eqZList(C(x1,x2),C(y1,y2)) -> and(eqZList(x1,y1),eqZList(x2,y2))
            eqZList(C(x1,x2),Z()) -> False()
            eqZList(Z(),C(y1,y2)) -> False()
            eqZList(Z(),Z()) -> True()
            first(C(x1,x2)) -> x1
            second(C(x1,x2)) -> x2
        - Weak TRS:
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1} / {C/2,False/0,True/0,Z/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,and,eqZList,first,second} and constructors {C,False
            ,True,Z}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
          a#(Z(),y,z) -> c_2()
          eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
          eqZList#(C(x1,x2),Z()) -> c_4()
          eqZList#(Z(),C(y1,y2)) -> c_5()
          eqZList#(Z(),Z()) -> c_6()
          first#(C(x1,x2)) -> c_7()
          second#(C(x1,x2)) -> c_8()
        Weak DPs
          and#(False(),False()) -> c_9()
          and#(False(),True()) -> c_10()
          and#(True(),False()) -> c_11()
          and#(True(),True()) -> c_12()
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
            a#(Z(),y,z) -> c_2()
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
            eqZList#(C(x1,x2),Z()) -> c_4()
            eqZList#(Z(),C(y1,y2)) -> c_5()
            eqZList#(Z(),Z()) -> c_6()
            first#(C(x1,x2)) -> c_7()
            second#(C(x1,x2)) -> c_8()
        - Weak DPs:
            and#(False(),False()) -> c_9()
            and#(False(),True()) -> c_10()
            and#(True(),False()) -> c_11()
            and#(True(),True()) -> c_12()
        - Weak TRS:
            a(C(x1,x2),y,z) -> C(a(x1,y,z),a(x2,y,y))
            a(Z(),y,z) -> Z()
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            eqZList(C(x1,x2),C(y1,y2)) -> and(eqZList(x1,y1),eqZList(x2,y2))
            eqZList(C(x1,x2),Z()) -> False()
            eqZList(Z(),C(y1,y2)) -> False()
            eqZList(Z(),Z()) -> True()
            first(C(x1,x2)) -> x1
            second(C(x1,x2)) -> x2
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/3,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,5,6,7,8}
        by application of
          Pre({2,4,5,6,7,8}) = {1,3}.
        Here rules are labelled as follows:
          1: a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
          2: a#(Z(),y,z) -> c_2()
          3: eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
          4: eqZList#(C(x1,x2),Z()) -> c_4()
          5: eqZList#(Z(),C(y1,y2)) -> c_5()
          6: eqZList#(Z(),Z()) -> c_6()
          7: first#(C(x1,x2)) -> c_7()
          8: second#(C(x1,x2)) -> c_8()
          9: and#(False(),False()) -> c_9()
          10: and#(False(),True()) -> c_10()
          11: and#(True(),False()) -> c_11()
          12: and#(True(),True()) -> c_12()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
        - Weak DPs:
            a#(Z(),y,z) -> c_2()
            and#(False(),False()) -> c_9()
            and#(False(),True()) -> c_10()
            and#(True(),False()) -> c_11()
            and#(True(),True()) -> c_12()
            eqZList#(C(x1,x2),Z()) -> c_4()
            eqZList#(Z(),C(y1,y2)) -> c_5()
            eqZList#(Z(),Z()) -> c_6()
            first#(C(x1,x2)) -> c_7()
            second#(C(x1,x2)) -> c_8()
        - Weak TRS:
            a(C(x1,x2),y,z) -> C(a(x1,y,z),a(x2,y,y))
            a(Z(),y,z) -> Z()
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            eqZList(C(x1,x2),C(y1,y2)) -> and(eqZList(x1,y1),eqZList(x2,y2))
            eqZList(C(x1,x2),Z()) -> False()
            eqZList(Z(),C(y1,y2)) -> False()
            eqZList(Z(),Z()) -> True()
            first(C(x1,x2)) -> x1
            second(C(x1,x2)) -> x2
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/3,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
             -->_2 a#(Z(),y,z) -> c_2():3
             -->_1 a#(Z(),y,z) -> c_2():3
             -->_2 a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y)):1
             -->_1 a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y)):1
          
          2:S:eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
             -->_3 eqZList#(Z(),Z()) -> c_6():10
             -->_2 eqZList#(Z(),Z()) -> c_6():10
             -->_3 eqZList#(Z(),C(y1,y2)) -> c_5():9
             -->_2 eqZList#(Z(),C(y1,y2)) -> c_5():9
             -->_3 eqZList#(C(x1,x2),Z()) -> c_4():8
             -->_2 eqZList#(C(x1,x2),Z()) -> c_4():8
             -->_1 and#(True(),True()) -> c_12():7
             -->_1 and#(True(),False()) -> c_11():6
             -->_1 and#(False(),True()) -> c_10():5
             -->_1 and#(False(),False()) -> c_9():4
             -->_3 eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2))
                                                     ,eqZList#(x1,y1)
                                                     ,eqZList#(x2,y2)):2
             -->_2 eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2))
                                                     ,eqZList#(x1,y1)
                                                     ,eqZList#(x2,y2)):2
          
          3:W:a#(Z(),y,z) -> c_2()
             
          
          4:W:and#(False(),False()) -> c_9()
             
          
          5:W:and#(False(),True()) -> c_10()
             
          
          6:W:and#(True(),False()) -> c_11()
             
          
          7:W:and#(True(),True()) -> c_12()
             
          
          8:W:eqZList#(C(x1,x2),Z()) -> c_4()
             
          
          9:W:eqZList#(Z(),C(y1,y2)) -> c_5()
             
          
          10:W:eqZList#(Z(),Z()) -> c_6()
             
          
          11:W:first#(C(x1,x2)) -> c_7()
             
          
          12:W:second#(C(x1,x2)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: second#(C(x1,x2)) -> c_8()
          11: first#(C(x1,x2)) -> c_7()
          4: and#(False(),False()) -> c_9()
          5: and#(False(),True()) -> c_10()
          6: and#(True(),False()) -> c_11()
          7: and#(True(),True()) -> c_12()
          8: eqZList#(C(x1,x2),Z()) -> c_4()
          9: eqZList#(Z(),C(y1,y2)) -> c_5()
          10: eqZList#(Z(),Z()) -> c_6()
          3: a#(Z(),y,z) -> c_2()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
        - Weak TRS:
            a(C(x1,x2),y,z) -> C(a(x1,y,z),a(x2,y,y))
            a(Z(),y,z) -> Z()
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            eqZList(C(x1,x2),C(y1,y2)) -> and(eqZList(x1,y1),eqZList(x2,y2))
            eqZList(C(x1,x2),Z()) -> False()
            eqZList(Z(),C(y1,y2)) -> False()
            eqZList(Z(),Z()) -> True()
            first(C(x1,x2)) -> x1
            second(C(x1,x2)) -> x2
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/3,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
             -->_2 a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y)):1
             -->_1 a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y)):1
          
          2:S:eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2)),eqZList#(x1,y1),eqZList#(x2,y2))
             -->_3 eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2))
                                                     ,eqZList#(x1,y1)
                                                     ,eqZList#(x2,y2)):2
             -->_2 eqZList#(C(x1,x2),C(y1,y2)) -> c_3(and#(eqZList(x1,y1),eqZList(x2,y2))
                                                     ,eqZList#(x1,y1)
                                                     ,eqZList#(x2,y2)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          eqZList#(C(x1,x2),C(y1,y2)) -> c_3(eqZList#(x1,y1),eqZList#(x2,y2))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(eqZList#(x1,y1),eqZList#(x2,y2))
        - Weak TRS:
            a(C(x1,x2),y,z) -> C(a(x1,y,z),a(x2,y,y))
            a(Z(),y,z) -> Z()
            and(False(),False()) -> False()
            and(False(),True()) -> False()
            and(True(),False()) -> False()
            and(True(),True()) -> True()
            eqZList(C(x1,x2),C(y1,y2)) -> and(eqZList(x1,y1),eqZList(x2,y2))
            eqZList(C(x1,x2),Z()) -> False()
            eqZList(Z(),C(y1,y2)) -> False()
            eqZList(Z(),Z()) -> True()
            first(C(x1,x2)) -> x1
            second(C(x1,x2)) -> x2
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
          eqZList#(C(x1,x2),C(y1,y2)) -> c_3(eqZList#(x1,y1),eqZList#(x2,y2))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(eqZList#(x1,y1),eqZList#(x2,y2))
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_3) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(C) = [1] x1 + [1] x2 + [8]
               p(False) = [2]                  
                p(True) = [1]                  
                   p(Z) = [1]                  
                   p(a) = [2] x1 + [8] x2 + [0]
                 p(and) = [1] x1 + [1] x2 + [0]
             p(eqZList) = [2] x1 + [2]         
               p(first) = [8] x1 + [4]         
              p(second) = [1] x1 + [1]         
                  p(a#) = [0]                  
                p(and#) = [2] x2 + [0]         
            p(eqZList#) = [2] x2 + [4]         
              p(first#) = [1] x1 + [4]         
             p(second#) = [1]                  
                 p(c_1) = [1] x1 + [1] x2 + [6]
                 p(c_2) = [1]                  
                 p(c_3) = [1] x1 + [1] x2 + [6]
                 p(c_4) = [1]                  
                 p(c_5) = [2]                  
                 p(c_6) = [8]                  
                 p(c_7) = [1]                  
                 p(c_8) = [0]                  
                 p(c_9) = [1]                  
                p(c_10) = [1]                  
                p(c_11) = [4]                  
                p(c_12) = [4]                  
          
          Following rules are strictly oriented:
          eqZList#(C(x1,x2),C(y1,y2)) = [2] y1 + [2] y2 + [20]              
                                      > [2] y1 + [2] y2 + [14]              
                                      = c_3(eqZList#(x1,y1),eqZList#(x2,y2))
          
          
          Following rules are (at-least) weakly oriented:
          a#(C(x1,x2),y,z) =  [0]                       
                           >= [6]                       
                           =  c_1(a#(x1,y,z),a#(x2,y,y))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
        - Weak DPs:
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(eqZList#(x1,y1),eqZList#(x2,y2))
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_1) = {1,2},
            uargs(c_3) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(C) = [1] x1 + [1] x2 + [8]         
               p(False) = [1]                           
                p(True) = [1]                           
                   p(Z) = [1]                           
                   p(a) = [1] x1 + [2] x2 + [2] x3 + [1]
                 p(and) = [2] x1 + [1] x2 + [1]         
             p(eqZList) = [2] x1 + [1] x2 + [1]         
               p(first) = [8]                           
              p(second) = [1]                           
                  p(a#) = [2] x1 + [1]                  
                p(and#) = [1] x1 + [8] x2 + [1]         
            p(eqZList#) = [1] x1 + [0]                  
              p(first#) = [1] x1 + [8]                  
             p(second#) = [1] x1 + [1]                  
                 p(c_1) = [1] x1 + [1] x2 + [14]        
                 p(c_2) = [0]                           
                 p(c_3) = [1] x1 + [1] x2 + [8]         
                 p(c_4) = [0]                           
                 p(c_5) = [2]                           
                 p(c_6) = [0]                           
                 p(c_7) = [0]                           
                 p(c_8) = [1]                           
                 p(c_9) = [0]                           
                p(c_10) = [1]                           
                p(c_11) = [4]                           
                p(c_12) = [0]                           
          
          Following rules are strictly oriented:
          a#(C(x1,x2),y,z) = [2] x1 + [2] x2 + [17]    
                           > [2] x1 + [2] x2 + [16]    
                           = c_1(a#(x1,y,z),a#(x2,y,y))
          
          
          Following rules are (at-least) weakly oriented:
          eqZList#(C(x1,x2),C(y1,y2)) =  [1] x1 + [1] x2 + [8]               
                                      >= [1] x1 + [1] x2 + [8]               
                                      =  c_3(eqZList#(x1,y1),eqZList#(x2,y2))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            a#(C(x1,x2),y,z) -> c_1(a#(x1,y,z),a#(x2,y,y))
            eqZList#(C(x1,x2),C(y1,y2)) -> c_3(eqZList#(x1,y1),eqZList#(x2,y2))
        - Signature:
            {a/3,and/2,eqZList/2,first/1,second/1,a#/3,and#/2,eqZList#/2,first#/1,second#/1} / {C/2,False/0,True/0,Z/0
            ,c_1/2,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,and#,eqZList#,first#,second#} and constructors {C
            ,False,True,Z}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))