WORST_CASE(?,O(n^1))
* Step 1: 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 2: DependencyPairs 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:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          0#() -> c_1()
          U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
          U12#(tt()) -> c_3()
          U21#(tt()) -> c_4()
          U31#(tt(),N) -> c_5(activate#(N))
          U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          activate#(X) -> c_8()
          activate#(n__0()) -> c_9(0#())
          activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
          activate#(n__s(X)) -> c_11(s#(X))
          isNat#(n__0()) -> c_12()
          isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
          isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
          plus#(X1,X2) -> c_15()
          s#(X) -> c_16()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            0#() -> c_1()
            U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
            U12#(tt()) -> c_3()
            U21#(tt()) -> c_4()
            U31#(tt(),N) -> c_5(activate#(N))
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            activate#(X) -> c_8()
            activate#(n__0()) -> c_9(0#())
            activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
            activate#(n__s(X)) -> c_11(s#(X))
            isNat#(n__0()) -> c_12()
            isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
            isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - 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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          0() -> n__0()
          U11(tt(),V2) -> U12(isNat(activate(V2)))
          U12(tt()) -> tt()
          U21(tt()) -> tt()
          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)
          0#() -> c_1()
          U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
          U12#(tt()) -> c_3()
          U21#(tt()) -> c_4()
          U31#(tt(),N) -> c_5(activate#(N))
          U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          activate#(X) -> c_8()
          activate#(n__0()) -> c_9(0#())
          activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
          activate#(n__s(X)) -> c_11(s#(X))
          isNat#(n__0()) -> c_12()
          isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
          isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
          plus#(X1,X2) -> c_15()
          s#(X) -> c_16()
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            0#() -> c_1()
            U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
            U12#(tt()) -> c_3()
            U21#(tt()) -> c_4()
            U31#(tt(),N) -> c_5(activate#(N))
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            activate#(X) -> c_8()
            activate#(n__0()) -> c_9(0#())
            activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
            activate#(n__s(X)) -> c_11(s#(X))
            isNat#(n__0()) -> c_12()
            isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
            isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - Strict TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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 = WgOnTrs}
    + 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(U11) = {1,2},
            uargs(U12) = {1},
            uargs(U21) = {1},
            uargs(isNat) = {1},
            uargs(plus) = {1,2},
            uargs(U11#) = {1,2},
            uargs(U12#) = {1},
            uargs(U21#) = {1},
            uargs(U42#) = {1,2,3},
            uargs(s#) = {1},
            uargs(c_2) = {1},
            uargs(c_5) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1},
            uargs(c_9) = {1},
            uargs(c_10) = {1},
            uargs(c_11) = {1},
            uargs(c_13) = {1},
            uargs(c_14) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [6]                           
                  p(U11) = [1] x1 + [1] x2 + [0]         
                  p(U12) = [1] x1 + [1]                  
                  p(U21) = [1] x1 + [2]                  
                  p(U31) = [0]                           
                  p(U41) = [0]                           
                  p(U42) = [0]                           
             p(activate) = [1] x1 + [2]                  
                p(isNat) = [1] x1 + [0]                  
                 p(n__0) = [5]                           
              p(n__plus) = [1] x1 + [1] x2 + [5]         
                 p(n__s) = [1] x1 + [5]                  
                 p(plus) = [1] x1 + [1] x2 + [6]         
                    p(s) = [1] x1 + [6]                  
                   p(tt) = [4]                           
                   p(0#) = [5]                           
                 p(U11#) = [1] x1 + [1] x2 + [0]         
                 p(U12#) = [1] x1 + [0]                  
                 p(U21#) = [1] x1 + [0]                  
                 p(U31#) = [4] x2 + [0]                  
                 p(U41#) = [1] x2 + [4] x3 + [0]         
                 p(U42#) = [1] x1 + [1] x2 + [1] x3 + [6]
            p(activate#) = [2] x1 + [3]                  
               p(isNat#) = [2] x1 + [1]                  
                p(plus#) = [2] x2 + [0]                  
                   p(s#) = [1] x1 + [0]                  
                  p(c_1) = [0]                           
                  p(c_2) = [1] x1 + [0]                  
                  p(c_3) = [0]                           
                  p(c_4) = [0]                           
                  p(c_5) = [1] x1 + [0]                  
                  p(c_6) = [1] x1 + [0]                  
                  p(c_7) = [1] x1 + [0]                  
                  p(c_8) = [0]                           
                  p(c_9) = [1] x1 + [0]                  
                 p(c_10) = [1] x1 + [0]                  
                 p(c_11) = [1] x1 + [0]                  
                 p(c_12) = [0]                           
                 p(c_13) = [1] x1 + [0]                  
                 p(c_14) = [1] x1 + [0]                  
                 p(c_15) = [0]                           
                 p(c_16) = [0]                           
          
          Following rules are strictly oriented:
                               0#() = [5]                                         
                                    > [0]                                         
                                    = c_1()                                       
          
                      U11#(tt(),V2) = [1] V2 + [4]                                
                                    > [1] V2 + [2]                                
                                    = c_2(U12#(isNat(activate(V2))))              
          
                         U12#(tt()) = [4]                                         
                                    > [0]                                         
                                    = c_3()                                       
          
                         U21#(tt()) = [4]                                         
                                    > [0]                                         
                                    = c_4()                                       
          
                       activate#(X) = [2] X + [3]                                 
                                    > [0]                                         
                                    = c_8()                                       
          
                  activate#(n__0()) = [13]                                        
                                    > [5]                                         
                                    = c_9(0#())                                   
          
          activate#(n__plus(X1,X2)) = [2] X1 + [2] X2 + [13]                      
                                    > [2] X2 + [0]                                
                                    = c_10(plus#(X1,X2))                          
          
                 activate#(n__s(X)) = [2] X + [13]                                
                                    > [1] X + [0]                                 
                                    = c_11(s#(X))                                 
          
                     isNat#(n__0()) = [11]                                        
                                    > [0]                                         
                                    = c_12()                                      
          
             isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [11]                      
                                    > [1] V1 + [1] V2 + [4]                       
                                    = c_13(U11#(isNat(activate(V1)),activate(V2)))
          
                   isNat#(n__s(V1)) = [2] V1 + [11]                               
                                    > [1] V1 + [2]                                
                                    = c_14(U21#(isNat(activate(V1))))             
          
                                0() = [6]                                         
                                    > [5]                                         
                                    = n__0()                                      
          
                       U11(tt(),V2) = [1] V2 + [4]                                
                                    > [1] V2 + [3]                                
                                    = U12(isNat(activate(V2)))                    
          
                          U12(tt()) = [5]                                         
                                    > [4]                                         
                                    = tt()                                        
          
                          U21(tt()) = [6]                                         
                                    > [4]                                         
                                    = tt()                                        
          
                        activate(X) = [1] X + [2]                                 
                                    > [1] X + [0]                                 
                                    = X                                           
          
                   activate(n__0()) = [7]                                         
                                    > [6]                                         
                                    = 0()                                         
          
           activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7]                       
                                    > [1] X1 + [1] X2 + [6]                       
                                    = plus(X1,X2)                                 
          
                  activate(n__s(X)) = [1] X + [7]                                 
                                    > [1] X + [6]                                 
                                    = s(X)                                        
          
                      isNat(n__0()) = [5]                                         
                                    > [4]                                         
                                    = tt()                                        
          
              isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]                       
                                    > [1] V1 + [1] V2 + [4]                       
                                    = U11(isNat(activate(V1)),activate(V2))       
          
                    isNat(n__s(V1)) = [1] V1 + [5]                                
                                    > [1] V1 + [4]                                
                                    = U21(isNat(activate(V1)))                    
          
                        plus(X1,X2) = [1] X1 + [1] X2 + [6]                       
                                    > [1] X1 + [1] X2 + [5]                       
                                    = n__plus(X1,X2)                              
          
                               s(X) = [1] X + [6]                                 
                                    > [1] X + [5]                                 
                                    = n__s(X)                                     
          
          
          Following rules are (at-least) weakly oriented:
            U31#(tt(),N) =  [4] N + [0]                                          
                         >= [2] N + [3]                                          
                         =  c_5(activate#(N))                                    
          
          U41#(tt(),M,N) =  [1] M + [4] N + [0]                                  
                         >= [1] M + [2] N + [12]                                 
                         =  c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          
          U42#(tt(),M,N) =  [1] M + [1] N + [10]                                 
                         >= [1] M + [1] N + [10]                                 
                         =  c_7(s#(plus(activate(N),activate(M))))               
          
            plus#(X1,X2) =  [2] X2 + [0]                                         
                         >= [0]                                                  
                         =  c_15()                                               
          
                   s#(X) =  [1] X + [0]                                          
                         >= [0]                                                  
                         =  c_16()                                               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U31#(tt(),N) -> c_5(activate#(N))
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - Weak DPs:
            0#() -> c_1()
            U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
            U12#(tt()) -> c_3()
            U21#(tt()) -> c_4()
            activate#(X) -> c_8()
            activate#(n__0()) -> c_9(0#())
            activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
            activate#(n__s(X)) -> c_11(s#(X))
            isNat#(n__0()) -> c_12()
            isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
            isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: U31#(tt(),N) -> c_5(activate#(N))
          2: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          3: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          4: plus#(X1,X2) -> c_15()
          5: s#(X) -> c_16()
          6: 0#() -> c_1()
          7: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
          8: U12#(tt()) -> c_3()
          9: U21#(tt()) -> c_4()
          10: activate#(X) -> c_8()
          11: activate#(n__0()) -> c_9(0#())
          12: activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
          13: activate#(n__s(X)) -> c_11(s#(X))
          14: isNat#(n__0()) -> c_12()
          15: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
          16: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - Weak DPs:
            0#() -> c_1()
            U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
            U12#(tt()) -> c_3()
            U21#(tt()) -> c_4()
            U31#(tt(),N) -> c_5(activate#(N))
            activate#(X) -> c_8()
            activate#(n__0()) -> c_9(0#())
            activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
            activate#(n__s(X)) -> c_11(s#(X))
            isNat#(n__0()) -> c_12()
            isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
            isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
             -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2
          
          2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
             -->_1 s#(X) -> c_16():4
          
          3:S:plus#(X1,X2) -> c_15()
             
          
          4:S:s#(X) -> c_16()
             
          
          5:W:0#() -> c_1()
             
          
          6:W:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
             -->_1 U12#(tt()) -> c_3():7
          
          7:W:U12#(tt()) -> c_3()
             
          
          8:W:U21#(tt()) -> c_4()
             
          
          9:W:U31#(tt(),N) -> c_5(activate#(N))
             -->_1 activate#(n__s(X)) -> c_11(s#(X)):13
             -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12
             -->_1 activate#(n__0()) -> c_9(0#()):11
             -->_1 activate#(X) -> c_8():10
          
          10:W:activate#(X) -> c_8()
             
          
          11:W:activate#(n__0()) -> c_9(0#())
             -->_1 0#() -> c_1():5
          
          12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
             -->_1 plus#(X1,X2) -> c_15():3
          
          13:W:activate#(n__s(X)) -> c_11(s#(X))
             -->_1 s#(X) -> c_16():4
          
          14:W:isNat#(n__0()) -> c_12()
             
          
          15:W:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
             -->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))):6
          
          16:W:isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
             -->_1 U21#(tt()) -> c_4():8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          16: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1))))
          15: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2)))
          14: isNat#(n__0()) -> c_12()
          10: activate#(X) -> c_8()
          11: activate#(n__0()) -> c_9(0#())
          8: U21#(tt()) -> c_4()
          6: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2))))
          7: U12#(tt()) -> c_3()
          5: 0#() -> c_1()
* Step 7: RemoveHeads WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - Weak DPs:
            U31#(tt(),N) -> c_5(activate#(N))
            activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
            activate#(n__s(X)) -> c_11(s#(X))
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
           -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2
        
        2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
           -->_1 s#(X) -> c_16():4
        
        3:S:plus#(X1,X2) -> c_15()
           
        
        4:S:s#(X) -> c_16()
           
        
        9:W:U31#(tt(),N) -> c_5(activate#(N))
           -->_1 activate#(n__s(X)) -> c_11(s#(X)):13
           -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12
        
        12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
           -->_1 plus#(X1,X2) -> c_15():3
        
        13:W:activate#(n__s(X)) -> c_11(s#(X))
           -->_1 s#(X) -> c_16():4
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(9,U31#(tt(),N) -> c_5(activate#(N)))]
* Step 8: RemoveHeads WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - Weak DPs:
            activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
            activate#(n__s(X)) -> c_11(s#(X))
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
           -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2
        
        2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
           -->_1 s#(X) -> c_16():4
        
        3:S:plus#(X1,X2) -> c_15()
           
        
        4:S:s#(X) -> c_16()
           
        
        12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))
           -->_1 plus#(X1,X2) -> c_15():3
        
        13:W:activate#(n__s(X)) -> c_11(s#(X))
           -->_1 s#(X) -> c_16():4
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(12,activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))),(13,activate#(n__s(X)) -> c_11(s#(X)))]
* Step 9: RemoveHeads WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            plus#(X1,X2) -> c_15()
            s#(X) -> c_16()
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
           -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2
        
        2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
           -->_1 s#(X) -> c_16():4
        
        3:S:plus#(X1,X2) -> c_15()
           
        
        4:S:s#(X) -> c_16()
           
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(3,plus#(X1,X2) -> c_15())]
* Step 10: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            s#(X) -> c_16()
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4}
        by application of
          Pre({4}) = {2}.
        Here rules are labelled as follows:
          1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          4: s#(X) -> c_16()
* Step 11: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
        - Weak DPs:
            s#(X) -> c_16()
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1}.
        Here rules are labelled as follows:
          1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          3: s#(X) -> c_16()
* Step 12: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
        - Weak DPs:
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            s#(X) -> c_16()
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          3: s#(X) -> c_16()
* Step 13: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
            U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
            s#(X) -> c_16()
        - Weak TRS:
            0() -> n__0()
            U11(tt(),V2) -> U12(isNat(activate(V2)))
            U12(tt()) -> tt()
            U21(tt()) -> tt()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
             -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2
          
          2:W:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
             -->_1 s#(X) -> c_16():3
          
          3:W:s#(X) -> c_16()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N)))
          2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M))))
          3: s#(X) -> c_16()
* Step 14: 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()
            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/1,0#/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,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0}
        - 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))