* Step 1: Sum WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__0()) -> tt()
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(N,0()) -> U31(isNat(N),N)
            plus(N,s(M)) -> U41(isNat(M),M,N)
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: InnermostRuleRemoval WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__0()) -> tt()
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(N,0()) -> U31(isNat(N),N)
            plus(N,s(M)) -> U41(isNat(M),M,N)
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          plus(N,0()) -> U31(isNat(N),N)
          plus(N,s(M)) -> U41(isNat(M),M,N)
        All above mentioned rules can be savely removed.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__0()) -> tt()
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                           
                 p(U11) = [1] x1 + [1] x2 + [0]         
                 p(U12) = [1] x1 + [0]                  
                 p(U21) = [1] x1 + [0]                  
                 p(U31) = [1] x2 + [0]                  
                 p(U41) = [1] x2 + [2] x3 + [0]         
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [9]
            p(activate) = [1] x1 + [0]                  
               p(isNat) = [1] x1 + [8]                  
                p(n__0) = [9]                           
             p(n__plus) = [1] x1 + [1] x2 + [0]         
                p(n__s) = [1] x1 + [0]                  
                p(plus) = [1] x1 + [1] x2 + [5]         
                   p(s) = [1] x1 + [12]                 
                  p(tt) = [8]                           
          
          Following rules are strictly oriented:
          activate(n__0()) = [9]                  
                           > [0]                  
                           = 0()                  
          
             isNat(n__0()) = [17]                 
                           > [8]                  
                           = tt()                 
          
               plus(X1,X2) = [1] X1 + [1] X2 + [5]
                           > [1] X1 + [1] X2 + [0]
                           = n__plus(X1,X2)       
          
                      s(X) = [1] X + [12]         
                           > [1] X + [0]          
                           = n__s(X)              
          
          
          Following rules are (at-least) weakly oriented:
                               0() =  [0]                                            
                                   >= [9]                                            
                                   =  n__0()                                         
          
                      U11(tt(),V2) =  [1] V2 + [8]                                   
                                   >= [1] V2 + [8]                                   
                                   =  U12(isNat(activate(V2)))                       
          
                         U12(tt()) =  [8]                                            
                                   >= [8]                                            
                                   =  tt()                                           
          
                         U21(tt()) =  [8]                                            
                                   >= [8]                                            
                                   =  tt()                                           
          
                       U31(tt(),N) =  [1] N + [0]                                    
                                   >= [1] N + [0]                                    
                                   =  activate(N)                                    
          
                     U41(tt(),M,N) =  [1] M + [2] N + [0]                            
                                   >= [1] M + [2] N + [17]                           
                                   =  U42(isNat(activate(N)),activate(M),activate(N))
          
                     U42(tt(),M,N) =  [1] M + [1] N + [17]                           
                                   >= [1] M + [1] N + [17]                           
                                   =  s(plus(activate(N),activate(M)))               
          
                       activate(X) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  X                                              
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [0]                          
                                   >= [1] X1 + [1] X2 + [5]                          
                                   =  plus(X1,X2)                                    
          
                 activate(n__s(X)) =  [1] X + [0]                                    
                                   >= [1] X + [12]                                   
                                   =  s(X)                                           
          
             isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [8]                          
                                   >= [1] V1 + [1] V2 + [8]                          
                                   =  U11(isNat(activate(V1)),activate(V2))          
          
                   isNat(n__s(V1)) =  [1] V1 + [8]                                   
                                   >= [1] V1 + [8]                                   
                                   =  U21(isNat(activate(V1)))                       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
        - Weak TRS:
            activate(n__0()) -> 0()
            isNat(n__0()) -> tt()
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [1]                            
                 p(U11) = [1] x1 + [1] x2 + [13]         
                 p(U12) = [1] x1 + [0]                   
                 p(U21) = [1] x1 + [0]                   
                 p(U31) = [1] x2 + [0]                   
                 p(U41) = [1] x2 + [2] x3 + [0]          
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [15]
            p(activate) = [1] x1 + [0]                   
               p(isNat) = [1] x1 + [2]                   
                p(n__0) = [1]                            
             p(n__plus) = [1] x1 + [1] x2 + [3]          
                p(n__s) = [1] x1 + [15]                  
                p(plus) = [1] x1 + [1] x2 + [3]          
                   p(s) = [1] x1 + [15]                  
                  p(tt) = [3]                            
          
          Following rules are strictly oriented:
             U11(tt(),V2) = [1] V2 + [16]           
                          > [1] V2 + [2]            
                          = U12(isNat(activate(V2)))
          
          isNat(n__s(V1)) = [1] V1 + [17]           
                          > [1] V1 + [2]            
                          = U21(isNat(activate(V1)))
          
          
          Following rules are (at-least) weakly oriented:
                               0() =  [1]                                            
                                   >= [1]                                            
                                   =  n__0()                                         
          
                         U12(tt()) =  [3]                                            
                                   >= [3]                                            
                                   =  tt()                                           
          
                         U21(tt()) =  [3]                                            
                                   >= [3]                                            
                                   =  tt()                                           
          
                       U31(tt(),N) =  [1] N + [0]                                    
                                   >= [1] N + [0]                                    
                                   =  activate(N)                                    
          
                     U41(tt(),M,N) =  [1] M + [2] N + [0]                            
                                   >= [1] M + [2] N + [17]                           
                                   =  U42(isNat(activate(N)),activate(M),activate(N))
          
                     U42(tt(),M,N) =  [1] M + [1] N + [18]                           
                                   >= [1] M + [1] N + [18]                           
                                   =  s(plus(activate(N),activate(M)))               
          
                       activate(X) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  X                                              
          
                  activate(n__0()) =  [1]                                            
                                   >= [1]                                            
                                   =  0()                                            
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [3]                          
                                   >= [1] X1 + [1] X2 + [3]                          
                                   =  plus(X1,X2)                                    
          
                 activate(n__s(X)) =  [1] X + [15]                                   
                                   >= [1] X + [15]                                   
                                   =  s(X)                                           
          
                     isNat(n__0()) =  [3]                                            
                                   >= [3]                                            
                                   =  tt()                                           
          
             isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [5]                          
                                   >= [1] V1 + [1] V2 + [15]                         
                                   =  U11(isNat(activate(V1)),activate(V2))          
          
                       plus(X1,X2) =  [1] X1 + [1] X2 + [3]                          
                                   >= [1] X1 + [1] X2 + [3]                          
                                   =  n__plus(X1,X2)                                 
          
                              s(X) =  [1] X + [15]                                   
                                   >= [1] X + [15]                                   
                                   =  n__s(X)                                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
        - Weak TRS:
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            activate(n__0()) -> 0()
            isNat(n__0()) -> tt()
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                           
                 p(U11) = [1] x1 + [1] x2 + [7]         
                 p(U12) = [1] x1 + [0]                  
                 p(U21) = [1] x1 + [1]                  
                 p(U31) = [2] x1 + [4] x2 + [0]         
                 p(U41) = [3] x2 + [2] x3 + [0]         
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [7]
            p(activate) = [1] x1 + [0]                  
               p(isNat) = [1] x1 + [2]                  
                p(n__0) = [2]                           
             p(n__plus) = [1] x1 + [1] x2 + [1]         
                p(n__s) = [1] x1 + [1]                  
                p(plus) = [1] x1 + [1] x2 + [1]         
                   p(s) = [1] x1 + [1]                  
                  p(tt) = [4]                           
          
          Following rules are strictly oriented:
              U21(tt()) = [5]                             
                        > [4]                             
                        = tt()                            
          
            U31(tt(),N) = [4] N + [8]                     
                        > [1] N + [0]                     
                        = activate(N)                     
          
          U42(tt(),M,N) = [1] M + [1] N + [11]            
                        > [1] M + [1] N + [2]             
                        = s(plus(activate(N),activate(M)))
          
          
          Following rules are (at-least) weakly oriented:
                               0() =  [0]                                            
                                   >= [2]                                            
                                   =  n__0()                                         
          
                      U11(tt(),V2) =  [1] V2 + [11]                                  
                                   >= [1] V2 + [2]                                   
                                   =  U12(isNat(activate(V2)))                       
          
                         U12(tt()) =  [4]                                            
                                   >= [4]                                            
                                   =  tt()                                           
          
                     U41(tt(),M,N) =  [3] M + [2] N + [0]                            
                                   >= [1] M + [2] N + [9]                            
                                   =  U42(isNat(activate(N)),activate(M),activate(N))
          
                       activate(X) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  X                                              
          
                  activate(n__0()) =  [2]                                            
                                   >= [0]                                            
                                   =  0()                                            
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [1]                          
                                   >= [1] X1 + [1] X2 + [1]                          
                                   =  plus(X1,X2)                                    
          
                 activate(n__s(X)) =  [1] X + [1]                                    
                                   >= [1] X + [1]                                    
                                   =  s(X)                                           
          
                     isNat(n__0()) =  [4]                                            
                                   >= [4]                                            
                                   =  tt()                                           
          
             isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [3]                          
                                   >= [1] V1 + [1] V2 + [9]                          
                                   =  U11(isNat(activate(V1)),activate(V2))          
          
                   isNat(n__s(V1)) =  [1] V1 + [3]                                   
                                   >= [1] V1 + [3]                                   
                                   =  U21(isNat(activate(V1)))                       
          
                       plus(X1,X2) =  [1] X1 + [1] X2 + [1]                          
                                   >= [1] X1 + [1] X2 + [1]                          
                                   =  n__plus(X1,X2)                                 
          
                              s(X) =  [1] X + [1]                                    
                                   >= [1] X + [1]                                    
                                   =  n__s(X)                                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            U12(tt()) -> tt()
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            activate(X) -> X
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
        - Weak TRS:
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(n__0()) -> 0()
            isNat(n__0()) -> tt()
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                           
                 p(U11) = [1] x1 + [1] x2 + [0]         
                 p(U12) = [1] x1 + [0]                  
                 p(U21) = [1] x1 + [1]                  
                 p(U31) = [4] x2 + [4]                  
                 p(U41) = [1] x2 + [2] x3 + [2]         
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [1]
            p(activate) = [1] x1 + [0]                  
               p(isNat) = [1] x1 + [0]                  
                p(n__0) = [0]                           
             p(n__plus) = [1] x1 + [1] x2 + [0]         
                p(n__s) = [1] x1 + [1]                  
                p(plus) = [1] x1 + [1] x2 + [0]         
                   p(s) = [1] x1 + [1]                  
                  p(tt) = [0]                           
          
          Following rules are strictly oriented:
          U41(tt(),M,N) = [1] M + [2] N + [2]                            
                        > [1] M + [2] N + [1]                            
                        = U42(isNat(activate(N)),activate(M),activate(N))
          
          
          Following rules are (at-least) weakly oriented:
                               0() =  [0]                                  
                                   >= [0]                                  
                                   =  n__0()                               
          
                      U11(tt(),V2) =  [1] V2 + [0]                         
                                   >= [1] V2 + [0]                         
                                   =  U12(isNat(activate(V2)))             
          
                         U12(tt()) =  [0]                                  
                                   >= [0]                                  
                                   =  tt()                                 
          
                         U21(tt()) =  [1]                                  
                                   >= [0]                                  
                                   =  tt()                                 
          
                       U31(tt(),N) =  [4] N + [4]                          
                                   >= [1] N + [0]                          
                                   =  activate(N)                          
          
                     U42(tt(),M,N) =  [1] M + [1] N + [1]                  
                                   >= [1] M + [1] N + [1]                  
                                   =  s(plus(activate(N),activate(M)))     
          
                       activate(X) =  [1] X + [0]                          
                                   >= [1] X + [0]                          
                                   =  X                                    
          
                  activate(n__0()) =  [0]                                  
                                   >= [0]                                  
                                   =  0()                                  
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [0]                
                                   >= [1] X1 + [1] X2 + [0]                
                                   =  plus(X1,X2)                          
          
                 activate(n__s(X)) =  [1] X + [1]                          
                                   >= [1] X + [1]                          
                                   =  s(X)                                 
          
                     isNat(n__0()) =  [0]                                  
                                   >= [0]                                  
                                   =  tt()                                 
          
             isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [0]                
                                   >= [1] V1 + [1] V2 + [0]                
                                   =  U11(isNat(activate(V1)),activate(V2))
          
                   isNat(n__s(V1)) =  [1] V1 + [1]                         
                                   >= [1] V1 + [1]                         
                                   =  U21(isNat(activate(V1)))             
          
                       plus(X1,X2) =  [1] X1 + [1] X2 + [0]                
                                   >= [1] X1 + [1] X2 + [0]                
                                   =  n__plus(X1,X2)                       
          
                              s(X) =  [1] X + [1]                          
                                   >= [1] X + [1]                          
                                   =  n__s(X)                              
          
        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 TRS:
            0() -> n__0()
            U12(tt()) -> tt()
            activate(X) -> X
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
        - Weak TRS:
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(n__0()) -> 0()
            isNat(n__0()) -> tt()
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                           
                 p(U11) = [1] x1 + [1] x2 + [0]         
                 p(U12) = [1] x1 + [1]                  
                 p(U21) = [1] x1 + [0]                  
                 p(U31) = [1] x2 + [2]                  
                 p(U41) = [2] x2 + [2] x3 + [3]         
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [3]
            p(activate) = [1] x1 + [0]                  
               p(isNat) = [1] x1 + [0]                  
                p(n__0) = [3]                           
             p(n__plus) = [1] x1 + [1] x2 + [0]         
                p(n__s) = [1] x1 + [1]                  
                p(plus) = [1] x1 + [1] x2 + [0]         
                   p(s) = [1] x1 + [1]                  
                  p(tt) = [2]                           
          
          Following rules are strictly oriented:
          U12(tt()) = [3] 
                    > [2] 
                    = tt()
          
          
          Following rules are (at-least) weakly oriented:
                               0() =  [0]                                            
                                   >= [3]                                            
                                   =  n__0()                                         
          
                      U11(tt(),V2) =  [1] V2 + [2]                                   
                                   >= [1] V2 + [1]                                   
                                   =  U12(isNat(activate(V2)))                       
          
                         U21(tt()) =  [2]                                            
                                   >= [2]                                            
                                   =  tt()                                           
          
                       U31(tt(),N) =  [1] N + [2]                                    
                                   >= [1] N + [0]                                    
                                   =  activate(N)                                    
          
                     U41(tt(),M,N) =  [2] M + [2] N + [3]                            
                                   >= [1] M + [2] N + [3]                            
                                   =  U42(isNat(activate(N)),activate(M),activate(N))
          
                     U42(tt(),M,N) =  [1] M + [1] N + [5]                            
                                   >= [1] M + [1] N + [1]                            
                                   =  s(plus(activate(N),activate(M)))               
          
                       activate(X) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  X                                              
          
                  activate(n__0()) =  [3]                                            
                                   >= [0]                                            
                                   =  0()                                            
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [0]                          
                                   >= [1] X1 + [1] X2 + [0]                          
                                   =  plus(X1,X2)                                    
          
                 activate(n__s(X)) =  [1] X + [1]                                    
                                   >= [1] X + [1]                                    
                                   =  s(X)                                           
          
                     isNat(n__0()) =  [3]                                            
                                   >= [2]                                            
                                   =  tt()                                           
          
             isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [0]                          
                                   >= [1] V1 + [1] V2 + [0]                          
                                   =  U11(isNat(activate(V1)),activate(V2))          
          
                   isNat(n__s(V1)) =  [1] V1 + [1]                                   
                                   >= [1] V1 + [0]                                   
                                   =  U21(isNat(activate(V1)))                       
          
                       plus(X1,X2) =  [1] X1 + [1] X2 + [0]                          
                                   >= [1] X1 + [1] X2 + [0]                          
                                   =  n__plus(X1,X2)                                 
          
                              s(X) =  [1] X + [1]                                    
                                   >= [1] X + [1]                                    
                                   =  n__s(X)                                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            activate(X) -> X
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
        - Weak TRS:
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(n__0()) -> 0()
            isNat(n__0()) -> tt()
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                           
                 p(U11) = [1] x1 + [1] x2 + [0]         
                 p(U12) = [1] x1 + [0]                  
                 p(U21) = [1] x1 + [0]                  
                 p(U31) = [1] x1 + [4] x2 + [7]         
                 p(U41) = [1] x2 + [4] x3 + [1]         
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [0]
            p(activate) = [1] x1 + [0]                  
               p(isNat) = [1] x1 + [0]                  
                p(n__0) = [5]                           
             p(n__plus) = [1] x1 + [1] x2 + [1]         
                p(n__s) = [1] x1 + [0]                  
                p(plus) = [1] x1 + [1] x2 + [1]         
                   p(s) = [1] x1 + [0]                  
                  p(tt) = [1]                           
          
          Following rules are strictly oriented:
          isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [1]                
                                > [1] V1 + [1] V2 + [0]                
                                = U11(isNat(activate(V1)),activate(V2))
          
          
          Following rules are (at-least) weakly oriented:
                               0() =  [0]                                            
                                   >= [5]                                            
                                   =  n__0()                                         
          
                      U11(tt(),V2) =  [1] V2 + [1]                                   
                                   >= [1] V2 + [0]                                   
                                   =  U12(isNat(activate(V2)))                       
          
                         U12(tt()) =  [1]                                            
                                   >= [1]                                            
                                   =  tt()                                           
          
                         U21(tt()) =  [1]                                            
                                   >= [1]                                            
                                   =  tt()                                           
          
                       U31(tt(),N) =  [4] N + [8]                                    
                                   >= [1] N + [0]                                    
                                   =  activate(N)                                    
          
                     U41(tt(),M,N) =  [1] M + [4] N + [1]                            
                                   >= [1] M + [2] N + [0]                            
                                   =  U42(isNat(activate(N)),activate(M),activate(N))
          
                     U42(tt(),M,N) =  [1] M + [1] N + [1]                            
                                   >= [1] M + [1] N + [1]                            
                                   =  s(plus(activate(N),activate(M)))               
          
                       activate(X) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  X                                              
          
                  activate(n__0()) =  [5]                                            
                                   >= [0]                                            
                                   =  0()                                            
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [1]                          
                                   >= [1] X1 + [1] X2 + [1]                          
                                   =  plus(X1,X2)                                    
          
                 activate(n__s(X)) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  s(X)                                           
          
                     isNat(n__0()) =  [5]                                            
                                   >= [1]                                            
                                   =  tt()                                           
          
                   isNat(n__s(V1)) =  [1] V1 + [0]                                   
                                   >= [1] V1 + [0]                                   
                                   =  U21(isNat(activate(V1)))                       
          
                       plus(X1,X2) =  [1] X1 + [1] X2 + [1]                          
                                   >= [1] X1 + [1] X2 + [1]                          
                                   =  n__plus(X1,X2)                                 
          
                              s(X) =  [1] X + [0]                                    
                                   >= [1] X + [0]                                    
                                   =  n__s(X)                                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 9: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            activate(X) -> X
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
        - Weak TRS:
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(n__0()) -> 0()
            isNat(n__0()) -> tt()
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                           
                 p(U11) = [1] x1 + [1] x2 + [0]         
                 p(U12) = [1] x1 + [0]                  
                 p(U21) = [1] x1 + [0]                  
                 p(U31) = [1] x2 + [1]                  
                 p(U41) = [1] x2 + [2] x3 + [7]         
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [1]
            p(activate) = [1] x1 + [1]                  
               p(isNat) = [1] x1 + [3]                  
                p(n__0) = [4]                           
             p(n__plus) = [1] x1 + [1] x2 + [2]         
                p(n__s) = [1] x1 + [1]                  
                p(plus) = [1] x1 + [1] x2 + [2]         
                   p(s) = [1] x1 + [1]                  
                  p(tt) = [4]                           
          
          Following rules are strictly oriented:
                       activate(X) = [1] X + [1]          
                                   > [1] X + [0]          
                                   = X                    
          
          activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3]
                                   > [1] X1 + [1] X2 + [2]
                                   = plus(X1,X2)          
          
                 activate(n__s(X)) = [1] X + [2]          
                                   > [1] X + [1]          
                                   = s(X)                 
          
          
          Following rules are (at-least) weakly oriented:
                            0() =  [0]                                            
                                >= [4]                                            
                                =  n__0()                                         
          
                   U11(tt(),V2) =  [1] V2 + [4]                                   
                                >= [1] V2 + [4]                                   
                                =  U12(isNat(activate(V2)))                       
          
                      U12(tt()) =  [4]                                            
                                >= [4]                                            
                                =  tt()                                           
          
                      U21(tt()) =  [4]                                            
                                >= [4]                                            
                                =  tt()                                           
          
                    U31(tt(),N) =  [1] N + [1]                                    
                                >= [1] N + [1]                                    
                                =  activate(N)                                    
          
                  U41(tt(),M,N) =  [1] M + [2] N + [7]                            
                                >= [1] M + [2] N + [7]                            
                                =  U42(isNat(activate(N)),activate(M),activate(N))
          
                  U42(tt(),M,N) =  [1] M + [1] N + [5]                            
                                >= [1] M + [1] N + [5]                            
                                =  s(plus(activate(N),activate(M)))               
          
               activate(n__0()) =  [5]                                            
                                >= [0]                                            
                                =  0()                                            
          
                  isNat(n__0()) =  [7]                                            
                                >= [4]                                            
                                =  tt()                                           
          
          isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [5]                          
                                >= [1] V1 + [1] V2 + [5]                          
                                =  U11(isNat(activate(V1)),activate(V2))          
          
                isNat(n__s(V1)) =  [1] V1 + [4]                                   
                                >= [1] V1 + [4]                                   
                                =  U21(isNat(activate(V1)))                       
          
                    plus(X1,X2) =  [1] X1 + [1] X2 + [2]                          
                                >= [1] X1 + [1] X2 + [2]                          
                                =  n__plus(X1,X2)                                 
          
                           s(X) =  [1] X + [1]                                    
                                >= [1] X + [1]                                    
                                =  n__s(X)                                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 10: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
        - Weak TRS:
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__0()) -> tt()
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(U42) = {1,2,3},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [4]                           
                 p(U11) = [1] x1 + [1] x2 + [1]         
                 p(U12) = [1] x1 + [1]                  
                 p(U21) = [1] x1 + [0]                  
                 p(U31) = [2] x2 + [2]                  
                 p(U41) = [1] x1 + [4] x2 + [4] x3 + [7]
                 p(U42) = [1] x1 + [1] x2 + [1] x3 + [6]
            p(activate) = [1] x1 + [1]                  
               p(isNat) = [1] x1 + [0]                  
                p(n__0) = [3]                           
             p(n__plus) = [1] x1 + [1] x2 + [3]         
                p(n__s) = [1] x1 + [1]                  
                p(plus) = [1] x1 + [1] x2 + [4]         
                   p(s) = [1] x1 + [1]                  
                  p(tt) = [2]                           
          
          Following rules are strictly oriented:
          0() = [4]   
              > [3]   
              = n__0()
          
          
          Following rules are (at-least) weakly oriented:
                      U11(tt(),V2) =  [1] V2 + [3]                                   
                                   >= [1] V2 + [2]                                   
                                   =  U12(isNat(activate(V2)))                       
          
                         U12(tt()) =  [3]                                            
                                   >= [2]                                            
                                   =  tt()                                           
          
                         U21(tt()) =  [2]                                            
                                   >= [2]                                            
                                   =  tt()                                           
          
                       U31(tt(),N) =  [2] N + [2]                                    
                                   >= [1] N + [1]                                    
                                   =  activate(N)                                    
          
                     U41(tt(),M,N) =  [4] M + [4] N + [9]                            
                                   >= [1] M + [2] N + [9]                            
                                   =  U42(isNat(activate(N)),activate(M),activate(N))
          
                     U42(tt(),M,N) =  [1] M + [1] N + [8]                            
                                   >= [1] M + [1] N + [7]                            
                                   =  s(plus(activate(N),activate(M)))               
          
                       activate(X) =  [1] X + [1]                                    
                                   >= [1] X + [0]                                    
                                   =  X                                              
          
                  activate(n__0()) =  [4]                                            
                                   >= [4]                                            
                                   =  0()                                            
          
          activate(n__plus(X1,X2)) =  [1] X1 + [1] X2 + [4]                          
                                   >= [1] X1 + [1] X2 + [4]                          
                                   =  plus(X1,X2)                                    
          
                 activate(n__s(X)) =  [1] X + [2]                                    
                                   >= [1] X + [1]                                    
                                   =  s(X)                                           
          
                     isNat(n__0()) =  [3]                                            
                                   >= [2]                                            
                                   =  tt()                                           
          
             isNat(n__plus(V1,V2)) =  [1] V1 + [1] V2 + [3]                          
                                   >= [1] V1 + [1] V2 + [3]                          
                                   =  U11(isNat(activate(V1)),activate(V2))          
          
                   isNat(n__s(V1)) =  [1] V1 + [1]                                   
                                   >= [1] V1 + [1]                                   
                                   =  U21(isNat(activate(V1)))                       
          
                       plus(X1,X2) =  [1] X1 + [1] X2 + [4]                          
                                   >= [1] X1 + [1] X2 + [3]                          
                                   =  n__plus(X1,X2)                                 
          
                              s(X) =  [1] X + [1]                                    
                                   >= [1] X + [1]                                    
                                   =  n__s(X)                                        
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 11: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            U31(tt(),N) -> activate(N)
            U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
            U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__plus(X1,X2)) -> plus(X1,X2)
            activate(n__s(X)) -> s(X)
            isNat(n__0()) -> tt()
            isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
            isNat(n__s(V1)) -> U21(isNat(activate(V1)))
            plus(X1,X2) -> n__plus(X1,X2)
            s(X) -> n__s(X)
        - Signature:
            {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus
            ,s} and constructors {n__0,n__plus,n__s,tt}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))