*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        and(tt(),X) -> activate(X)
        fst(pair(X,Y)) -> X
        head(cons(N,XS)) -> N
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        sel(N,XS) -> head(afterNth(N,XS))
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
        tail(cons(N,XS)) -> activate(XS)
        take(N,XS) -> fst(splitAt(N,XS))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0}
      Obligation:
        Innermost
        basic terms: {U11,U12,activate,afterNth,and,fst,head,natsFrom,sel,snd,splitAt,tail,take}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        activate#(X) -> c_3()
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        and#(tt(),X) -> c_6(activate#(X))
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        tail#(cons(N,XS)) -> c_15(activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        activate#(X) -> c_3()
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        and#(tt(),X) -> c_6(activate#(X))
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        tail#(cons(N,XS)) -> c_15(activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        and(tt(),X) -> activate(X)
        fst(pair(X,Y)) -> X
        head(cons(N,XS)) -> N
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        sel(N,XS) -> head(afterNth(N,XS))
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
        tail(cons(N,XS)) -> activate(XS)
        take(N,XS) -> fst(splitAt(N,XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        activate#(X) -> c_3()
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        and#(tt(),X) -> c_6(activate#(X))
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        tail#(cons(N,XS)) -> c_15(activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        activate#(X) -> c_3()
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        and#(tt(),X) -> c_6(activate#(X))
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        tail#(cons(N,XS)) -> c_15(activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {3,7,8,9,10,12,13}
      by application of
        Pre({3,7,8,9,10,12,13}) = {1,2,4,5,6,11,14,15,16}.
      Here rules are labelled as follows:
        1:  U11#(tt(),N,X,XS) ->              
              c_1(U12#(splitAt(activate(N)    
                              ,activate(XS))  
                      ,activate(X))           
                 ,splitAt#(activate(N)        
                          ,activate(XS))      
                 ,activate#(N)                
                 ,activate#(XS)               
                 ,activate#(X))               
        2:  U12#(pair(YS,ZS),X) ->            
              c_2(activate#(X))               
        3:  activate#(X) -> c_3()             
        4:  activate#(n__natsFrom(X)) ->      
              c_4(natsFrom#(X))               
        5:  afterNth#(N,XS) ->                
              c_5(snd#(splitAt(N,XS))         
                 ,splitAt#(N,XS))             
        6:  and#(tt(),X) ->                   
              c_6(activate#(X))               
        7:  fst#(pair(X,Y)) -> c_7()          
        8:  head#(cons(N,XS)) -> c_8()        
        9:  natsFrom#(N) -> c_9()             
        10: natsFrom#(X) -> c_10()            
        11: sel#(N,XS) ->                     
              c_11(head#(afterNth(N,XS))      
                  ,afterNth#(N,XS))           
        12: snd#(pair(X,Y)) -> c_12()         
        13: splitAt#(0(),XS) -> c_13()        
        14: splitAt#(s(N),cons(X,XS)) ->      
              c_14(U11#(tt(),N,X,activate(XS))
                  ,activate#(XS))             
        15: tail#(cons(N,XS)) ->              
              c_15(activate#(XS))             
        16: take#(N,XS) ->                    
              c_16(fst#(splitAt(N,XS))        
                  ,splitAt#(N,XS))            
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        and#(tt(),X) -> c_6(activate#(X))
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        tail#(cons(N,XS)) -> c_15(activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        activate#(X) -> c_3()
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {3}
      by application of
        Pre({3}) = {1,2,5,7,8}.
      Here rules are labelled as follows:
        1:  U11#(tt(),N,X,XS) ->              
              c_1(U12#(splitAt(activate(N)    
                              ,activate(XS))  
                      ,activate(X))           
                 ,splitAt#(activate(N)        
                          ,activate(XS))      
                 ,activate#(N)                
                 ,activate#(XS)               
                 ,activate#(X))               
        2:  U12#(pair(YS,ZS),X) ->            
              c_2(activate#(X))               
        3:  activate#(n__natsFrom(X)) ->      
              c_4(natsFrom#(X))               
        4:  afterNth#(N,XS) ->                
              c_5(snd#(splitAt(N,XS))         
                 ,splitAt#(N,XS))             
        5:  and#(tt(),X) ->                   
              c_6(activate#(X))               
        6:  sel#(N,XS) ->                     
              c_11(head#(afterNth(N,XS))      
                  ,afterNth#(N,XS))           
        7:  splitAt#(s(N),cons(X,XS)) ->      
              c_14(U11#(tt(),N,X,activate(XS))
                  ,activate#(XS))             
        8:  tail#(cons(N,XS)) ->              
              c_15(activate#(XS))             
        9:  take#(N,XS) ->                    
              c_16(fst#(splitAt(N,XS))        
                  ,splitAt#(N,XS))            
        10: activate#(X) -> c_3()             
        11: fst#(pair(X,Y)) -> c_7()          
        12: head#(cons(N,XS)) -> c_8()        
        13: natsFrom#(N) -> c_9()             
        14: natsFrom#(X) -> c_10()            
        15: snd#(pair(X,Y)) -> c_12()         
        16: splitAt#(0(),XS) -> c_13()        
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        and#(tt(),X) -> c_6(activate#(X))
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        tail#(cons(N,XS)) -> c_15(activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        activate#(X) -> c_3()
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {2,4,7}
      by application of
        Pre({2,4,7}) = {1}.
      Here rules are labelled as follows:
        1:  U11#(tt(),N,X,XS) ->              
              c_1(U12#(splitAt(activate(N)    
                              ,activate(XS))  
                      ,activate(X))           
                 ,splitAt#(activate(N)        
                          ,activate(XS))      
                 ,activate#(N)                
                 ,activate#(XS)               
                 ,activate#(X))               
        2:  U12#(pair(YS,ZS),X) ->            
              c_2(activate#(X))               
        3:  afterNth#(N,XS) ->                
              c_5(snd#(splitAt(N,XS))         
                 ,splitAt#(N,XS))             
        4:  and#(tt(),X) ->                   
              c_6(activate#(X))               
        5:  sel#(N,XS) ->                     
              c_11(head#(afterNth(N,XS))      
                  ,afterNth#(N,XS))           
        6:  splitAt#(s(N),cons(X,XS)) ->      
              c_14(U11#(tt(),N,X,activate(XS))
                  ,activate#(XS))             
        7:  tail#(cons(N,XS)) ->              
              c_15(activate#(XS))             
        8:  take#(N,XS) ->                    
              c_16(fst#(splitAt(N,XS))        
                  ,splitAt#(N,XS))            
        9:  activate#(X) -> c_3()             
        10: activate#(n__natsFrom(X)) ->      
              c_4(natsFrom#(X))               
        11: fst#(pair(X,Y)) -> c_7()          
        12: head#(cons(N,XS)) -> c_8()        
        13: natsFrom#(N) -> c_9()             
        14: natsFrom#(X) -> c_10()            
        15: snd#(pair(X,Y)) -> c_12()         
        16: splitAt#(0(),XS) -> c_13()        
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        U12#(pair(YS,ZS),X) -> c_2(activate#(X))
        activate#(X) -> c_3()
        activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
        and#(tt(),X) -> c_6(activate#(X))
        fst#(pair(X,Y)) -> c_7()
        head#(cons(N,XS)) -> c_8()
        natsFrom#(N) -> c_9()
        natsFrom#(X) -> c_10()
        snd#(pair(X,Y)) -> c_12()
        splitAt#(0(),XS) -> c_13()
        tail#(cons(N,XS)) -> c_15(activate#(XS))
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
           -->_5 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_4 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_3 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_1 U12#(pair(YS,ZS),X) -> c_2(activate#(X)):6
           -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
           -->_2 splitAt#(0(),XS) -> c_13():15
           -->_5 activate#(X) -> c_3():7
           -->_4 activate#(X) -> c_3():7
           -->_3 activate#(X) -> c_3():7
        
        2:S:afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
           -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
           -->_2 splitAt#(0(),XS) -> c_13():15
           -->_1 snd#(pair(X,Y)) -> c_12():14
        
        3:S:sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
           -->_1 head#(cons(N,XS)) -> c_8():11
           -->_2 afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)):2
        
        4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
           -->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_2 activate#(X) -> c_3():7
           -->_1 U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X)):1
        
        5:S:take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
           -->_2 splitAt#(0(),XS) -> c_13():15
           -->_1 fst#(pair(X,Y)) -> c_7():10
           -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
        
        6:W:U12#(pair(YS,ZS),X) -> c_2(activate#(X))
           -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_1 activate#(X) -> c_3():7
        
        7:W:activate#(X) -> c_3()
           
        
        8:W:activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
           -->_1 natsFrom#(X) -> c_10():13
           -->_1 natsFrom#(N) -> c_9():12
        
        9:W:and#(tt(),X) -> c_6(activate#(X))
           -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_1 activate#(X) -> c_3():7
        
        10:W:fst#(pair(X,Y)) -> c_7()
           
        
        11:W:head#(cons(N,XS)) -> c_8()
           
        
        12:W:natsFrom#(N) -> c_9()
           
        
        13:W:natsFrom#(X) -> c_10()
           
        
        14:W:snd#(pair(X,Y)) -> c_12()
           
        
        15:W:splitAt#(0(),XS) -> c_13()
           
        
        16:W:tail#(cons(N,XS)) -> c_15(activate#(XS))
           -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
           -->_1 activate#(X) -> c_3():7
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        16: tail#(cons(N,XS)) ->        
              c_15(activate#(XS))       
        9:  and#(tt(),X) ->             
              c_6(activate#(X))         
        10: fst#(pair(X,Y)) -> c_7()    
        11: head#(cons(N,XS)) -> c_8()  
        14: snd#(pair(X,Y)) -> c_12()   
        15: splitAt#(0(),XS) -> c_13()  
        6:  U12#(pair(YS,ZS),X) ->      
              c_2(activate#(X))         
        7:  activate#(X) -> c_3()       
        8:  activate#(n__natsFrom(X)) ->
              c_4(natsFrom#(X))         
        12: natsFrom#(N) -> c_9()       
        13: natsFrom#(X) -> c_10()      
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
        afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
        sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
        take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      SimplifyRHS
    Proof:
      Consider the dependency graph
        1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X))
           -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
        
        2:S:afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
           -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
        
        3:S:sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
           -->_2 afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)):2
        
        4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
           -->_1 U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),splitAt#(activate(N),activate(XS)),activate#(N),activate#(XS),activate#(X)):1
        
        5:S:take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
           -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
        
      Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
        U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        afterNth#(N,XS) -> c_5(splitAt#(N,XS))
        sel#(N,XS) -> c_11(afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        take#(N,XS) -> c_16(splitAt#(N,XS))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        afterNth#(N,XS) -> c_5(splitAt#(N,XS))
        sel#(N,XS) -> c_11(afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        take#(N,XS) -> c_16(splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
        U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        afterNth(N,XS) -> snd(splitAt(N,XS))
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        snd(pair(X,Y)) -> Y
        splitAt(0(),XS) -> pair(nil(),XS)
        splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
        U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        afterNth#(N,XS) -> c_5(splitAt#(N,XS))
        sel#(N,XS) -> c_11(afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        take#(N,XS) -> c_16(splitAt#(N,XS))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        afterNth#(N,XS) -> c_5(splitAt#(N,XS))
        sel#(N,XS) -> c_11(afterNth#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        take#(N,XS) -> c_16(splitAt#(N,XS))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      RemoveHeads
    Proof:
      Consider the dependency graph
      
      1:S:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
         -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
      
      2:S:afterNth#(N,XS) -> c_5(splitAt#(N,XS))
         -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
      
      3:S:sel#(N,XS) -> c_11(afterNth#(N,XS))
         -->_1 afterNth#(N,XS) -> c_5(splitAt#(N,XS)):2
      
      4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
         -->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1
      
      5:S:take#(N,XS) -> c_16(splitAt#(N,XS))
         -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):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).
      
      [(3,sel#(N,XS) -> c_11(afterNth#(N,XS))),(5,take#(N,XS) -> c_16(splitAt#(N,XS)))]
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        afterNth#(N,XS) -> c_5(splitAt#(N,XS))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      RemoveHeads
    Proof:
      Consider the dependency graph
      
      1:S:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
         -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
      
      2:S:afterNth#(N,XS) -> c_5(splitAt#(N,XS))
         -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
      
      4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
         -->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1
      
      
      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).
      
      [(2,afterNth#(N,XS) -> c_5(splitAt#(N,XS)))]
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        activate(X) -> X
        activate(n__natsFrom(X)) -> natsFrom(X)
        natsFrom(N) -> cons(N,n__natsFrom(s(N)))
        natsFrom(X) -> n__natsFrom(X)
      Signature:
        {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
      Obligation:
        Innermost
        basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
    Applied Processor:
      PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        1: U11#(tt(),N,X,XS) ->         
             c_1(splitAt#(activate(N)   
                         ,activate(XS)))
        
      Consider the set of all dependency pairs
        1: U11#(tt(),N,X,XS) ->         
             c_1(splitAt#(activate(N)   
                         ,activate(XS)))
        4: splitAt#(s(N),cons(X,XS)) -> 
             c_14(U11#(tt()             
                      ,N                
                      ,X                
                      ,activate(XS)))   
      Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
      SPACE(?,?)on application of the dependency pairs
        {1}
      These cover all (indirect) predecessors of dependency pairs
        {1,4}
      their number of applications is equally bounded.
      The dependency pairs are shifted into the weak component.
  *** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
          splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          activate(X) -> X
          activate(n__natsFrom(X)) -> natsFrom(X)
          natsFrom(N) -> cons(N,n__natsFrom(s(N)))
          natsFrom(X) -> n__natsFrom(X)
        Signature:
          {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
        Obligation:
          Innermost
          basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
      Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {activate,natsFrom,U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}
        TcT has computed the following interpretation:
                    p(0) = [2]                           
                  p(U11) = [8] x2 + [1] x3 + [8] x4 + [1]
                  p(U12) = [2] x1 + [1] x2 + [0]         
             p(activate) = [1] x1 + [0]                  
             p(afterNth) = [1] x1 + [1] x2 + [1]         
                  p(and) = [2] x2 + [2]                  
                 p(cons) = [0]                           
                  p(fst) = [2] x1 + [1]                  
                 p(head) = [1] x1 + [1]                  
          p(n__natsFrom) = [0]                           
             p(natsFrom) = [0]                           
                  p(nil) = [1]                           
                 p(pair) = [1] x1 + [2]                  
                    p(s) = [1] x1 + [4]                  
                  p(sel) = [4] x1 + [1] x2 + [1]         
                  p(snd) = [0]                           
              p(splitAt) = [1] x1 + [1] x2 + [1]         
                 p(tail) = [4] x1 + [0]                  
                 p(take) = [2] x2 + [1]                  
                   p(tt) = [2]                           
                 p(U11#) = [6] x1 + [2] x2 + [4]         
                 p(U12#) = [8] x1 + [2]                  
            p(activate#) = [1] x1 + [2]                  
            p(afterNth#) = [2]                           
                 p(and#) = [1] x1 + [0]                  
                 p(fst#) = [4]                           
                p(head#) = [1] x1 + [0]                  
            p(natsFrom#) = [1] x1 + [2]                  
                 p(sel#) = [1]                           
                 p(snd#) = [0]                           
             p(splitAt#) = [2] x1 + [9]                  
                p(tail#) = [1] x1 + [1]                  
                p(take#) = [1]                           
                  p(c_1) = [1] x1 + [4]                  
                  p(c_2) = [2]                           
                  p(c_3) = [1]                           
                  p(c_4) = [1] x1 + [0]                  
                  p(c_5) = [8]                           
                  p(c_6) = [1] x1 + [1]                  
                  p(c_7) = [0]                           
                  p(c_8) = [0]                           
                  p(c_9) = [2]                           
                 p(c_10) = [1]                           
                 p(c_11) = [4] x1 + [4]                  
                 p(c_12) = [1]                           
                 p(c_13) = [4]                           
                 p(c_14) = [1] x1 + [1]                  
                 p(c_15) = [1] x1 + [2]                  
                 p(c_16) = [1]                           
        
        Following rules are strictly oriented:
        U11#(tt(),N,X,XS) = [2] N + [16]               
                          > [2] N + [13]               
                          = c_1(splitAt#(activate(N)   
                                        ,activate(XS)))
        
        
        Following rules are (at-least) weakly oriented:
        splitAt#(s(N),cons(X,XS)) =  [2] N + [17]             
                                  >= [2] N + [17]             
                                  =  c_14(U11#(tt()           
                                              ,N              
                                              ,X              
                                              ,activate(XS))) 
        
                      activate(X) =  [1] X + [0]              
                                  >= [1] X + [0]              
                                  =  X                        
        
         activate(n__natsFrom(X)) =  [0]                      
                                  >= [0]                      
                                  =  natsFrom(X)              
        
                      natsFrom(N) =  [0]                      
                                  >= [0]                      
                                  =  cons(N,n__natsFrom(s(N)))
        
                      natsFrom(X) =  [0]                      
                                  >= [0]                      
                                  =  n__natsFrom(X)           
        
  *** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        Strict TRS Rules:
          
        Weak DP Rules:
          U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
        Weak TRS Rules:
          activate(X) -> X
          activate(n__natsFrom(X)) -> natsFrom(X)
          natsFrom(N) -> cons(N,n__natsFrom(s(N)))
          natsFrom(X) -> n__natsFrom(X)
        Signature:
          {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
        Obligation:
          Innermost
          basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
          splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
        Weak TRS Rules:
          activate(X) -> X
          activate(n__natsFrom(X)) -> natsFrom(X)
          natsFrom(N) -> cons(N,n__natsFrom(s(N)))
          natsFrom(X) -> n__natsFrom(X)
        Signature:
          {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
        Obligation:
          Innermost
          basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
             -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):2
          
          2:W:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
             -->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: U11#(tt(),N,X,XS) ->         
               c_1(splitAt#(activate(N)   
                           ,activate(XS)))
          2: splitAt#(s(N),cons(X,XS)) -> 
               c_14(U11#(tt()             
                        ,N                
                        ,X                
                        ,activate(XS)))   
  *** 1.1.1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          activate(X) -> X
          activate(n__natsFrom(X)) -> natsFrom(X)
          natsFrom(N) -> cons(N,n__natsFrom(s(N)))
          natsFrom(X) -> n__natsFrom(X)
        Signature:
          {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
        Obligation:
          Innermost
          basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).