*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
        activate(X) -> X
        activate(n__cons(X1,X2)) -> cons(X1,X2)
        activate(n__from(X)) -> from(X)
        cons(X1,X2) -> n__cons(X1,X2)
        from(X) -> cons(X,n__from(s(X)))
        from(X) -> n__from(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1}
      Obligation:
        Full
        basic terms: {2nd,activate,cons,from}/{n__cons,n__from,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y))
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        activate#(n__from(X)) -> c_4(from#(X))
        cons#(X1,X2) -> c_5(X1,X2)
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y))
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        activate#(n__from(X)) -> c_4(from#(X))
        cons#(X1,X2) -> c_5(X1,X2)
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
      Strict TRS Rules:
        2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
        activate(X) -> X
        activate(n__cons(X1,X2)) -> cons(X1,X2)
        activate(n__from(X)) -> from(X)
        cons(X1,X2) -> n__cons(X1,X2)
        from(X) -> cons(X,n__from(s(X)))
        from(X) -> n__from(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
      Obligation:
        Full
        basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        activate#(n__from(X)) -> c_4(from#(X))
        cons#(X1,X2) -> c_5(X1,X2)
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        activate#(n__from(X)) -> c_4(from#(X))
        cons#(X1,X2) -> c_5(X1,X2)
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
      Obligation:
        Full
        basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                p(2nd) = [0]         
           p(activate) = [0]         
               p(cons) = [0]         
               p(from) = [0]         
            p(n__cons) = [0]         
            p(n__from) = [0]         
                  p(s) = [0]         
               p(2nd#) = [0]         
          p(activate#) = [1]         
              p(cons#) = [0]         
              p(from#) = [5]         
                p(c_1) = [0]         
                p(c_2) = [0]         
                p(c_3) = [1] x1 + [0]
                p(c_4) = [1] x1 + [0]
                p(c_5) = [0]         
                p(c_6) = [1] x1 + [0]
                p(c_7) = [0]         
        
        Following rules are strictly oriented:
                     activate#(X) = [1]                        
                                  > [0]                        
                                  = c_2(X)                     
        
        activate#(n__cons(X1,X2)) = [1]                        
                                  > [0]                        
                                  = c_3(cons#(X1,X2))          
        
                         from#(X) = [5]                        
                                  > [0]                        
                                  = c_6(cons#(X,n__from(s(X))))
        
                         from#(X) = [5]                        
                                  > [0]                        
                                  = c_7(X)                     
        
        
        Following rules are (at-least) weakly oriented:
        activate#(n__from(X)) =  [1]          
                              >= [5]          
                              =  c_4(from#(X))
        
                 cons#(X1,X2) =  [0]          
                              >= [0]          
                              =  c_5(X1,X2)   
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(n__from(X)) -> c_4(from#(X))
        cons#(X1,X2) -> c_5(X1,X2)
      Strict TRS Rules:
        
      Weak DP Rules:
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
      Weak TRS Rules:
        
      Signature:
        {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
      Obligation:
        Full
        basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                p(2nd) = [0]         
           p(activate) = [0]         
               p(cons) = [0]         
               p(from) = [0]         
            p(n__cons) = [0]         
            p(n__from) = [0]         
                  p(s) = [0]         
               p(2nd#) = [0]         
          p(activate#) = [11]        
              p(cons#) = [11]        
              p(from#) = [11]        
                p(c_1) = [0]         
                p(c_2) = [0]         
                p(c_3) = [1] x1 + [0]
                p(c_4) = [1] x1 + [0]
                p(c_5) = [0]         
                p(c_6) = [1] x1 + [0]
                p(c_7) = [1]         
        
        Following rules are strictly oriented:
        cons#(X1,X2) = [11]      
                     > [0]       
                     = c_5(X1,X2)
        
        
        Following rules are (at-least) weakly oriented:
                     activate#(X) =  [11]                       
                                  >= [0]                        
                                  =  c_2(X)                     
        
        activate#(n__cons(X1,X2)) =  [11]                       
                                  >= [11]                       
                                  =  c_3(cons#(X1,X2))          
        
            activate#(n__from(X)) =  [11]                       
                                  >= [11]                       
                                  =  c_4(from#(X))              
        
                         from#(X) =  [11]                       
                                  >= [11]                       
                                  =  c_6(cons#(X,n__from(s(X))))
        
                         from#(X) =  [11]                       
                                  >= [1]                        
                                  =  c_7(X)                     
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(n__from(X)) -> c_4(from#(X))
      Strict TRS Rules:
        
      Weak DP Rules:
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        cons#(X1,X2) -> c_5(X1,X2)
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
      Weak TRS Rules:
        
      Signature:
        {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
      Obligation:
        Full
        basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                p(2nd) = [0]          
           p(activate) = [0]          
               p(cons) = [0]          
               p(from) = [0]          
            p(n__cons) = [0]          
            p(n__from) = [2]          
                  p(s) = [0]          
               p(2nd#) = [0]          
          p(activate#) = [1] x1 + [11]
              p(cons#) = [0]          
              p(from#) = [0]          
                p(c_1) = [0]          
                p(c_2) = [0]          
                p(c_3) = [1] x1 + [11]
                p(c_4) = [1] x1 + [0] 
                p(c_5) = [0]          
                p(c_6) = [1] x1 + [0] 
                p(c_7) = [0]          
        
        Following rules are strictly oriented:
        activate#(n__from(X)) = [13]         
                              > [0]          
                              = c_4(from#(X))
        
        
        Following rules are (at-least) weakly oriented:
                     activate#(X) =  [1] X + [11]               
                                  >= [0]                        
                                  =  c_2(X)                     
        
        activate#(n__cons(X1,X2)) =  [11]                       
                                  >= [11]                       
                                  =  c_3(cons#(X1,X2))          
        
                     cons#(X1,X2) =  [0]                        
                                  >= [0]                        
                                  =  c_5(X1,X2)                 
        
                         from#(X) =  [0]                        
                                  >= [0]                        
                                  =  c_6(cons#(X,n__from(s(X))))
        
                         from#(X) =  [0]                        
                                  >= [0]                        
                                  =  c_7(X)                     
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        activate#(X) -> c_2(X)
        activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
        activate#(n__from(X)) -> c_4(from#(X))
        cons#(X1,X2) -> c_5(X1,X2)
        from#(X) -> c_6(cons#(X,n__from(s(X))))
        from#(X) -> c_7(X)
      Weak TRS Rules:
        
      Signature:
        {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
      Obligation:
        Full
        basic terms: {2nd#,activate#,cons#,from#}/{n__cons,n__from,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).