WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                             ,admit#(carry(x,u,v),z))
          admit#(x,nil()) -> c_2()
          cond#(true(),y) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                               ,admit#(carry(x,u,v),z))
            admit#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3()
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/2,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3}
        by application of
          Pre({2,3}) = {1}.
        Here rules are labelled as follows:
          1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                ,admit#(carry(x,u,v),z))
          2: admit#(x,nil()) -> c_2()
          3: cond#(true(),y) -> c_3()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                               ,admit#(carry(x,u,v),z))
        - Weak DPs:
            admit#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3()
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/2,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                 ,admit#(carry(x,u,v),z))
             -->_2 admit#(x,nil()) -> c_2():2
             -->_2 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                      ,admit#(carry(x,u,v),z)):1
          
          2:W:admit#(x,nil()) -> c_2()
             
          
          3:W:cond#(true(),y) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: cond#(true(),y) -> c_3()
          2: admit#(x,nil()) -> c_2()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                               ,admit#(carry(x,u,v),z))
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/2,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                 ,admit#(carry(x,u,v),z))
             -->_2 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                      ,admit#(carry(x,u,v),z)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + 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}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(.) = [1] x1 + [1] x2 + [0]         
                 p(=) = [1] x1 + [1] x2 + [1]         
             p(admit) = [1] x2 + [1]                  
             p(carry) = [1] x1 + [1] x2 + [1] x3 + [0]
              p(cond) = [1] x1 + [1] x2 + [1]         
               p(nil) = [1]                           
               p(sum) = [1] x1 + [2]                  
              p(true) = [1]                           
                 p(w) = [2]                           
            p(admit#) = [10] x1 + [10] x2 + [3]       
             p(cond#) = [1] x2 + [8]                  
               p(c_1) = [1] x1 + [3]                  
               p(c_2) = [0]                           
               p(c_3) = [1]                           
          
          Following rules are strictly oriented:
          admit#(x,.(u,.(v,.(w(),z)))) = [10] u + [10] v + [10] x + [10] z + [23]
                                       > [10] u + [10] v + [10] x + [10] z + [6] 
                                       = c_1(admit#(carry(x,u,v),z))             
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))