WORST_CASE(?,O(n^1))
* Step 1: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(.) = {2},
            uargs(cond) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x2 + [0]                  
                p(=) = [1] x1 + [1] x2 + [0]         
            p(admit) = [0]                           
            p(carry) = [1] x1 + [1] x2 + [1] x3 + [0]
             p(cond) = [1] x2 + [13]                 
              p(nil) = [4]                           
              p(sum) = [1] x1 + [1] x2 + [1] x3 + [0]
             p(true) = [0]                           
                p(w) = [0]                           
          
          Following rules are strictly oriented:
          cond(true(),y) = [1] y + [13]
                         > [1] y + [0] 
                         = y           
          
          
          Following rules are (at-least) weakly oriented:
          admit(x,.(u,.(v,.(w(),z)))) =  [0]                                                           
                                      >= [13]                                                          
                                      =  cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
          
                       admit(x,nil()) =  [0]                                                           
                                      >= [4]                                                           
                                      =  nil()                                                         
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap 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()
        - Weak TRS:
            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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(.) = {2},
            uargs(cond) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x2 + [2]
                p(=) = [1] x1 + [2]
            p(admit) = [5]         
            p(carry) = [1] x3 + [1]
             p(cond) = [1] x2 + [4]
              p(nil) = [1]         
              p(sum) = [1] x2 + [8]
             p(true) = [4]         
                p(w) = [1]         
          
          Following rules are strictly oriented:
          admit(x,nil()) = [5]  
                         > [1]  
                         = nil()
          
          
          Following rules are (at-least) weakly oriented:
          admit(x,.(u,.(v,.(w(),z)))) =  [5]                                                           
                                      >= [15]                                                          
                                      =  cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
          
                       cond(true(),y) =  [1] y + [4]                                                   
                                      >= [1] y + [0]                                                   
                                      =  y                                                             
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap 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)))))
        - Weak TRS:
            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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(.) = {2},
            uargs(cond) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x1 + [1] x2 + [0] 
                p(=) = [10]                  
            p(admit) = [1] x1 + [2] x2 + [10]
            p(carry) = [1] x3 + [0]          
             p(cond) = [1] x2 + [2]          
              p(nil) = [3]                   
              p(sum) = [1] x1 + [0]          
             p(true) = [8]                   
                p(w) = [4]                   
          
          Following rules are strictly oriented:
          admit(x,.(u,.(v,.(w(),z)))) = [2] u + [2] v + [1] x + [2] z + [18]                          
                                      > [1] u + [2] v + [2] z + [16]                                  
                                      = cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
          
          
          Following rules are (at-least) weakly oriented:
          admit(x,nil()) =  [1] x + [16]
                         >= [3]         
                         =  nil()       
          
          cond(true(),y) =  [1] y + [2] 
                         >= [1] y + [0] 
                         =  y           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - 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} / {./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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))