*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        activate(X) -> X
        activate(n__first(X1,X2)) -> first(X1,X2)
        activate(n__from(X)) -> from(X)
        first(X1,X2) -> n__first(X1,X2)
        first(0(),X) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        from(X) -> cons(X,n__from(s(X)))
        from(X) -> n__from(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
      Obligation:
        Innermost
        basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                 p(0) = [0]         
          p(activate) = [1] x1 + [4]
              p(cons) = [1] x2 + [0]
             p(first) = [1] x2 + [0]
              p(from) = [0]         
          p(n__first) = [1] x2 + [0]
           p(n__from) = [0]         
               p(nil) = [0]         
                 p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
                      activate(X) = [1] X + [4] 
                                  > [1] X + [0] 
                                  = X           
        
        activate(n__first(X1,X2)) = [1] X2 + [4]
                                  > [1] X2 + [0]
                                  = first(X1,X2)
        
             activate(n__from(X)) = [4]         
                                  > [0]         
                                  = from(X)     
        
        
        Following rules are (at-least) weakly oriented:
                 first(X1,X2) =  [1] X2 + [0]                   
                              >= [1] X2 + [0]                   
                              =  n__first(X1,X2)                
        
                 first(0(),X) =  [1] X + [0]                    
                              >= [0]                            
                              =  nil()                          
        
        first(s(X),cons(Y,Z)) =  [1] Z + [0]                    
                              >= [1] Z + [4]                    
                              =  cons(Y,n__first(X,activate(Z)))
        
                      from(X) =  [0]                            
                              >= [0]                            
                              =  cons(X,n__from(s(X)))          
        
                      from(X) =  [0]                            
                              >= [0]                            
                              =  n__from(X)                     
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        first(X1,X2) -> n__first(X1,X2)
        first(0(),X) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        from(X) -> cons(X,n__from(s(X)))
        from(X) -> n__from(X)
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__first(X1,X2)) -> first(X1,X2)
        activate(n__from(X)) -> from(X)
      Signature:
        {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
      Obligation:
        Innermost
        basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                 p(0) = [0]                   
          p(activate) = [10] x1 + [8]         
              p(cons) = [1] x2 + [0]          
             p(first) = [8] x1 + [10] x2 + [0]
              p(from) = [2]                   
          p(n__first) = [1] x1 + [1] x2 + [1] 
           p(n__from) = [0]                   
               p(nil) = [0]                   
                 p(s) = [1] x1 + [2]          
        
        Following rules are strictly oriented:
        first(s(X),cons(Y,Z)) = [8] X + [10] Z + [16]          
                              > [1] X + [10] Z + [9]           
                              = cons(Y,n__first(X,activate(Z)))
        
                      from(X) = [2]                            
                              > [0]                            
                              = cons(X,n__from(s(X)))          
        
                      from(X) = [2]                            
                              > [0]                            
                              = n__from(X)                     
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  [10] X + [8]            
                                  >= [1] X + [0]             
                                  =  X                       
        
        activate(n__first(X1,X2)) =  [10] X1 + [10] X2 + [18]
                                  >= [8] X1 + [10] X2 + [0]  
                                  =  first(X1,X2)            
        
             activate(n__from(X)) =  [8]                     
                                  >= [2]                     
                                  =  from(X)                 
        
                     first(X1,X2) =  [8] X1 + [10] X2 + [0]  
                                  >= [1] X1 + [1] X2 + [1]   
                                  =  n__first(X1,X2)         
        
                     first(0(),X) =  [10] X + [0]            
                                  >= [0]                     
                                  =  nil()                   
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        first(X1,X2) -> n__first(X1,X2)
        first(0(),X) -> nil()
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__first(X1,X2)) -> first(X1,X2)
        activate(n__from(X)) -> from(X)
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        from(X) -> cons(X,n__from(s(X)))
        from(X) -> n__from(X)
      Signature:
        {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
      Obligation:
        Innermost
        basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                 p(0) = [2]         
          p(activate) = [2] x1 + [8]
              p(cons) = [1] x2 + [1]
             p(first) = [2] x2 + [7]
              p(from) = [4]         
          p(n__first) = [1] x2 + [0]
           p(n__from) = [0]         
               p(nil) = [1]         
                 p(s) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        first(X1,X2) = [2] X2 + [7]   
                     > [1] X2 + [0]   
                     = n__first(X1,X2)
        
        first(0(),X) = [2] X + [7]    
                     > [1]            
                     = nil()          
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  [2] X + [8]                    
                                  >= [1] X + [0]                    
                                  =  X                              
        
        activate(n__first(X1,X2)) =  [2] X2 + [8]                   
                                  >= [2] X2 + [7]                   
                                  =  first(X1,X2)                   
        
             activate(n__from(X)) =  [8]                            
                                  >= [4]                            
                                  =  from(X)                        
        
            first(s(X),cons(Y,Z)) =  [2] Z + [9]                    
                                  >= [2] Z + [9]                    
                                  =  cons(Y,n__first(X,activate(Z)))
        
                          from(X) =  [4]                            
                                  >= [1]                            
                                  =  cons(X,n__from(s(X)))          
        
                          from(X) =  [4]                            
                                  >= [0]                            
                                  =  n__from(X)                     
        
      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:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__first(X1,X2)) -> first(X1,X2)
        activate(n__from(X)) -> from(X)
        first(X1,X2) -> n__first(X1,X2)
        first(0(),X) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        from(X) -> cons(X,n__from(s(X)))
        from(X) -> n__from(X)
      Signature:
        {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
      Obligation:
        Innermost
        basic terms: {activate,first,from}/{0,cons,n__first,n__from,nil,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).