*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
      Obligation:
        Full
        basic terms: {admit,cond}/{.,=,carry,nil,sum,true,w}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      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#(x,nil()) -> c_2()
        cond#(true(),y) -> c_3(y)
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
        admit#(x,nil()) -> c_2()
        cond#(true(),y) -> c_3(y)
      Strict TRS Rules:
        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
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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_3/1}
      Obligation:
        Full
        basic terms: {admit#,cond#}/{.,=,carry,nil,sum,true,w}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        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()
        admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
        admit#(x,nil()) -> c_2()
        cond#(true(),y) -> c_3(y)
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
        admit#(x,nil()) -> c_2()
        cond#(true(),y) -> c_3(y)
      Strict TRS Rules:
        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 DP Rules:
        
      Weak TRS Rules:
        
      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_3/1}
      Obligation:
        Full
        basic terms: {admit#,cond#}/{.,=,carry,nil,sum,true,w}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,2}
      by application of
        Pre({1,2}) = {3}.
      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))))))
        2: admit#(x,nil()) -> c_2()                    
        3: cond#(true(),y) -> c_3(y)                   
*** 1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        cond#(true(),y) -> c_3(y)
      Strict TRS Rules:
        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 DP Rules:
        admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
        admit#(x,nil()) -> c_2()
      Weak TRS Rules:
        
      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_3/1}
      Obligation:
        Full
        basic terms: {admit#,cond#}/{.,=,carry,nil,sum,true,w}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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},
          uargs(cond#) = {2},
          uargs(c_1) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
               p(.) = [1] x2 + [0]         
               p(=) = [2]                  
           p(admit) = [0]                  
           p(carry) = [1] x3 + [0]         
            p(cond) = [4] x1 + [1] x2 + [4]
             p(nil) = [1]                  
             p(sum) = [1] x2 + [1]         
            p(true) = [0]                  
               p(w) = [0]                  
          p(admit#) = [4]                  
           p(cond#) = [1] x2 + [1]         
             p(c_1) = [1] x1 + [3]         
             p(c_2) = [0]                  
             p(c_3) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        cond#(true(),y) = [1] y + [1]
                        > [1] y + [0]
                        = c_3(y)     
        
        
        Following rules are (at-least) weakly oriented:
        admit#(x,.(u,.(v,.(w(),z)))) =  [4]                                       
                                     >= [4]                                       
                                     =  c_1(cond#(=(sum(x,u,v),w())               
                                                 ,.(u                             
                                                   ,.(v                           
                                                     ,.(w()                       
                                                       ,admit(carry(x,u,v),z))))))
        
                     admit#(x,nil()) =  [4]                                       
                                     >= [0]                                       
                                     =  c_2()                                     
        
         admit(x,.(u,.(v,.(w(),z)))) =  [0]                                       
                                     >= [12]                                      
                                     =  cond(=(sum(x,u,v),w())                    
                                            ,.(u                                  
                                              ,.(v                                
                                                ,.(w(),admit(carry(x,u,v),z)))))  
        
                      admit(x,nil()) =  [0]                                       
                                     >= [1]                                       
                                     =  nil()                                     
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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 DP Rules:
        admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
        admit#(x,nil()) -> c_2()
        cond#(true(),y) -> c_3(y)
      Weak TRS Rules:
        
      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_3/1}
      Obligation:
        Full
        basic terms: {admit#,cond#}/{.,=,carry,nil,sum,true,w}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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},
          uargs(cond#) = {2},
          uargs(c_1) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
               p(.) = [1] x1 + [1] x2 + [0]
               p(=) = [1] x1 + [0]         
           p(admit) = [2] x2 + [0]         
           p(carry) = [1] x1 + [1] x3 + [1]
            p(cond) = [1] x2 + [1]         
             p(nil) = [4]                  
             p(sum) = [1] x1 + [1] x3 + [1]
            p(true) = [3]                  
               p(w) = [4]                  
          p(admit#) = [1] x1 + [2] x2 + [4]
           p(cond#) = [1] x1 + [1] x2 + [0]
             p(c_1) = [1] x1 + [0]         
             p(c_2) = [1]                  
             p(c_3) = [1] x1 + [3]         
        
        Following rules are strictly oriented:
        admit(x,.(u,.(v,.(w(),z)))) = [2] u + [2] v + [2] z + [8]             
                                    > [1] u + [1] v + [2] z + [5]             
                                    = cond(=(sum(x,u,v),w())                  
                                          ,.(u                                
                                            ,.(v                              
                                              ,.(w(),admit(carry(x,u,v),z)))))
        
                     admit(x,nil()) = [8]                                     
                                    > [4]                                     
                                    = nil()                                   
        
        
        Following rules are (at-least) weakly oriented:
        admit#(x,.(u,.(v,.(w(),z)))) =  [2] u + [2] v + [1] x + [2] z + [12]      
                                     >= [1] u + [2] v + [1] x + [2] z + [5]       
                                     =  c_1(cond#(=(sum(x,u,v),w())               
                                                 ,.(u                             
                                                   ,.(v                           
                                                     ,.(w()                       
                                                       ,admit(carry(x,u,v),z))))))
        
                     admit#(x,nil()) =  [1] x + [12]                              
                                     >= [1]                                       
                                     =  c_2()                                     
        
                     cond#(true(),y) =  [1] y + [3]                               
                                     >= [1] y + [3]                               
                                     =  c_3(y)                                    
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
        admit#(x,nil()) -> c_2()
        cond#(true(),y) -> c_3(y)
      Weak TRS Rules:
        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()
      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_3/1}
      Obligation:
        Full
        basic terms: {admit#,cond#}/{.,=,carry,nil,sum,true,w}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).