MAYBE

Problem:
 terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
 sqr(0()) -> 0()
 sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
 dbl(0()) -> 0()
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 add(0(),X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 first(0(),X) -> nil()
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 terms(X) -> n__terms(X)
 add(X1,X2) -> n__add(X1,X2)
 s(X) -> n__s(X)
 dbl(X) -> n__dbl(X)
 first(X1,X2) -> n__first(X1,X2)
 activate(n__terms(X)) -> terms(X)
 activate(n__add(X1,X2)) -> add(X1,X2)
 activate(n__s(X)) -> s(X)
 activate(n__dbl(X)) -> dbl(X)
 activate(n__first(X1,X2)) -> first(X1,X2)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   terms#(N) -> s#(N)
   terms#(N) -> sqr#(N)
   sqr#(s(X)) -> dbl#(activate(X))
   sqr#(s(X)) -> activate#(X)
   sqr#(s(X)) -> sqr#(activate(X))
   sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
   dbl#(s(X)) -> activate#(X)
   dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
   add#(s(X),Y) -> activate#(X)
   add#(s(X),Y) -> s#(n__add(activate(X),Y))
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   first#(s(X),cons(Y,Z)) -> activate#(X)
   activate#(n__terms(X)) -> terms#(X)
   activate#(n__add(X1,X2)) -> add#(X1,X2)
   activate#(n__s(X)) -> s#(X)
   activate#(n__dbl(X)) -> dbl#(X)
   activate#(n__first(X1,X2)) -> first#(X1,X2)
  TRS:
   terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
   sqr(0()) -> 0()
   sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
   dbl(0()) -> 0()
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0(),X) -> X
   add(s(X),Y) -> s(n__add(activate(X),Y))
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
   terms(X) -> n__terms(X)
   add(X1,X2) -> n__add(X1,X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1,X2) -> n__first(X1,X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1,X2)) -> add(X1,X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1,X2)) -> first(X1,X2)
   activate(X) -> X
  TDG Processor:
   DPs:
    terms#(N) -> s#(N)
    terms#(N) -> sqr#(N)
    sqr#(s(X)) -> dbl#(activate(X))
    sqr#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X))
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    dbl#(s(X)) -> activate#(X)
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    add#(s(X),Y) -> activate#(X)
    add#(s(X),Y) -> s#(n__add(activate(X),Y))
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    activate#(n__s(X)) -> s#(X)
    activate#(n__dbl(X)) -> dbl#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2)
   TRS:
    terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
    sqr(0()) -> 0()
    sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
    dbl(0()) -> 0()
    dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
    add(0(),X) -> X
    add(s(X),Y) -> s(n__add(activate(X),Y))
    first(0(),X) -> nil()
    first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
    terms(X) -> n__terms(X)
    add(X1,X2) -> n__add(X1,X2)
    s(X) -> n__s(X)
    dbl(X) -> n__dbl(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__terms(X)) -> terms(X)
    activate(n__add(X1,X2)) -> add(X1,X2)
    activate(n__s(X)) -> s(X)
    activate(n__dbl(X)) -> dbl(X)
    activate(n__first(X1,X2)) -> first(X1,X2)
    activate(X) -> X
   graph:
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__dbl(X)) -> dbl#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(X)) -> terms#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__dbl(X)) -> dbl#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    add#(s(X),Y) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__dbl(X)) -> dbl#(X) ->
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    activate#(n__dbl(X)) -> dbl#(X) ->
    dbl#(s(X)) -> activate#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> s#(n__add(activate(X),Y))
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> activate#(X)
    activate#(n__terms(X)) -> terms#(X) -> terms#(N) -> sqr#(N)
    activate#(n__terms(X)) -> terms#(X) -> terms#(N) -> s#(N)
    dbl#(s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    dbl#(s(X)) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    dbl#(s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    dbl#(s(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
    dbl#(s(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(X)
    sqr#(s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    sqr#(s(X)) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    sqr#(s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    sqr#(s(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
    sqr#(s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    sqr#(s(X)) -> dbl#(activate(X)) ->
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    sqr#(s(X)) -> dbl#(activate(X)) -> dbl#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X)) ->
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    sqr#(s(X)) -> sqr#(activate(X)) ->
    sqr#(s(X)) -> sqr#(activate(X))
    sqr#(s(X)) -> sqr#(activate(X)) -> sqr#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X)) -> sqr#(s(X)) -> dbl#(activate(X))
    terms#(N) -> sqr#(N) ->
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(activate(X))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> activate#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(activate(X))
   SCC Processor:
    #sccs: 1
    #rules: 12
    #arcs: 43/289
    DPs:
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     activate#(n__terms(X)) -> terms#(X)
     terms#(N) -> sqr#(N)
     sqr#(s(X)) -> dbl#(activate(X))
     dbl#(s(X)) -> activate#(X)
     activate#(n__add(X1,X2)) -> add#(X1,X2)
     add#(s(X),Y) -> activate#(X)
     activate#(n__dbl(X)) -> dbl#(X)
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     first#(s(X),cons(Y,Z)) -> activate#(X)
     sqr#(s(X)) -> activate#(X)
     sqr#(s(X)) -> sqr#(activate(X))
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
     add(0(),X) -> X
     add(s(X),Y) -> s(n__add(activate(X),Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
     terms(X) -> n__terms(X)
     add(X1,X2) -> n__add(X1,X2)
     s(X) -> n__s(X)
     dbl(X) -> n__dbl(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__add(X1,X2)) -> add(X1,X2)
     activate(n__s(X)) -> s(X)
     activate(n__dbl(X)) -> dbl(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [first#](x0, x1) = x0 + x1 + 0,
      
      [add#](x0, x1) = x0 + x1 + 0,
      
      [activate#](x0) = x0 + 0,
      
      [dbl#](x0) = x0 + 0,
      
      [sqr#](x0) = 1x0 + 1,
      
      [terms#](x0) = 2x0 + 1,
      
      [n__first](x0, x1) = x0 + 4x1 + 0,
      
      [nil] = 0,
      
      [first](x0, x1) = x0 + 4x1 + 0,
      
      [add](x0, x1) = x0 + x1 + 0,
      
      [n__s](x0) = x0,
      
      [n__dbl](x0) = x0 + 0,
      
      [n__add](x0, x1) = x0 + x1,
      
      [dbl](x0) = x0 + 0,
      
      [activate](x0) = x0 + 0,
      
      [0] = 0,
      
      [cons](x0, x1) = x0 + x1 + 0,
      
      [n__terms](x0) = 4x0 + 1,
      
      [s](x0) = x0,
      
      [recip](x0) = x0 + 0,
      
      [sqr](x0) = x0 + 0,
      
      [terms](x0) = 4x0 + 1
     orientation:
      first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z + 0 = activate#(Z)
      
      activate#(n__terms(X)) = 4X + 1 >= 2X + 1 = terms#(X)
      
      terms#(N) = 2N + 1 >= 1N + 1 = sqr#(N)
      
      sqr#(s(X)) = 1X + 1 >= X + 0 = dbl#(activate(X))
      
      dbl#(s(X)) = X + 0 >= X + 0 = activate#(X)
      
      activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add#(X1,X2)
      
      add#(s(X),Y) = X + Y + 0 >= X + 0 = activate#(X)
      
      activate#(n__dbl(X)) = X + 0 >= X + 0 = dbl#(X)
      
      activate#(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 + X2 + 0 = first#(X1,X2)
      
      first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + 0 = activate#(X)
      
      sqr#(s(X)) = 1X + 1 >= X + 0 = activate#(X)
      
      sqr#(s(X)) = 1X + 1 >= 1X + 1 = sqr#(activate(X))
      
      terms(N) = 4N + 1 >= 4N + 1 = cons(recip(sqr(N)),n__terms(s(N)))
      
      sqr(0()) = 0 >= 0 = 0()
      
      sqr(s(X)) = X + 0 >= X + 0 = s(n__add(sqr(activate(X)),dbl(activate(X))))
      
      dbl(0()) = 0 >= 0 = 0()
      
      dbl(s(X)) = X + 0 >= X + 0 = s(n__s(n__dbl(activate(X))))
      
      add(0(),X) = X + 0 >= X = X
      
      add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(n__add(activate(X),Y))
      
      first(0(),X) = 4X + 0 >= 0 = nil()
      
      first(s(X),cons(Y,Z)) = X + 4Y + 4Z + 4 >= X + Y + 4Z + 4 = cons(Y,n__first(activate(X),activate(Z)))
      
      terms(X) = 4X + 1 >= 4X + 1 = n__terms(X)
      
      add(X1,X2) = X1 + X2 + 0 >= X1 + X2 = n__add(X1,X2)
      
      s(X) = X >= X = n__s(X)
      
      dbl(X) = X + 0 >= X + 0 = n__dbl(X)
      
      first(X1,X2) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = n__first(X1,X2)
      
      activate(n__terms(X)) = 4X + 1 >= 4X + 1 = terms(X)
      
      activate(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add(X1,X2)
      
      activate(n__s(X)) = X + 0 >= X = s(X)
      
      activate(n__dbl(X)) = X + 0 >= X + 0 = dbl(X)
      
      activate(n__first(X1,X2)) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = first(X1,X2)
      
      activate(X) = X + 0 >= X = X
     problem:
      DPs:
       first#(s(X),cons(Y,Z)) -> activate#(Z)
       activate#(n__terms(X)) -> terms#(X)
       terms#(N) -> sqr#(N)
       dbl#(s(X)) -> activate#(X)
       activate#(n__add(X1,X2)) -> add#(X1,X2)
       add#(s(X),Y) -> activate#(X)
       activate#(n__dbl(X)) -> dbl#(X)
       activate#(n__first(X1,X2)) -> first#(X1,X2)
       first#(s(X),cons(Y,Z)) -> activate#(X)
       sqr#(s(X)) -> sqr#(activate(X))
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
       add(0(),X) -> X
       add(s(X),Y) -> s(n__add(activate(X),Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
       terms(X) -> n__terms(X)
       add(X1,X2) -> n__add(X1,X2)
       s(X) -> n__s(X)
       dbl(X) -> n__dbl(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__add(X1,X2)) -> add(X1,X2)
       activate(n__s(X)) -> s(X)
       activate(n__dbl(X)) -> dbl(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
     SCC Processor:
      #sccs: 2
      #rules: 8
      #arcs: 32/100
      DPs:
       first#(s(X),cons(Y,Z)) -> activate#(Z)
       activate#(n__add(X1,X2)) -> add#(X1,X2)
       add#(s(X),Y) -> activate#(X)
       activate#(n__dbl(X)) -> dbl#(X)
       dbl#(s(X)) -> activate#(X)
       activate#(n__first(X1,X2)) -> first#(X1,X2)
       first#(s(X),cons(Y,Z)) -> activate#(X)
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
       add(0(),X) -> X
       add(s(X),Y) -> s(n__add(activate(X),Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
       terms(X) -> n__terms(X)
       add(X1,X2) -> n__add(X1,X2)
       s(X) -> n__s(X)
       dbl(X) -> n__dbl(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__add(X1,X2)) -> add(X1,X2)
       activate(n__s(X)) -> s(X)
       activate(n__dbl(X)) -> dbl(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
      Size-Change Termination Processor:
       DPs:
        
       TRS:
        terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
        sqr(0()) -> 0()
        sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
        dbl(0()) -> 0()
        dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
        add(0(),X) -> X
        add(s(X),Y) -> s(n__add(activate(X),Y))
        first(0(),X) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
        terms(X) -> n__terms(X)
        add(X1,X2) -> n__add(X1,X2)
        s(X) -> n__s(X)
        dbl(X) -> n__dbl(X)
        first(X1,X2) -> n__first(X1,X2)
        activate(n__terms(X)) -> terms(X)
        activate(n__add(X1,X2)) -> add(X1,X2)
        activate(n__s(X)) -> s(X)
        activate(n__dbl(X)) -> dbl(X)
        activate(n__first(X1,X2)) -> first(X1,X2)
        activate(X) -> X
       The DP: first#(s(X),cons(Y,Z)) -> activate#(Z) has the edges:
         1 >   0
       The DP: activate#(n__add(X1,X2)) -> add#(X1,X2) has the edges:
         0 >   1
         0 >   0
       The DP: add#(s(X),Y) -> activate#(X) has the edges:
         0 >   0
       The DP: activate#(n__dbl(X)) -> dbl#(X) has the edges:
         0 >   0
       The DP: dbl#(s(X)) -> activate#(X) has the edges:
         0 >   0
       The DP: activate#(n__first(X1,X2)) -> first#(X1,X2) has the edges:
         0 >   1
         0 >   0
       The DP: first#(s(X),cons(Y,Z)) -> activate#(X) has the edges:
         0 >   0
       Qed
      
      DPs:
       sqr#(s(X)) -> sqr#(activate(X))
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
       add(0(),X) -> X
       add(s(X),Y) -> s(n__add(activate(X),Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
       terms(X) -> n__terms(X)
       add(X1,X2) -> n__add(X1,X2)
       s(X) -> n__s(X)
       dbl(X) -> n__dbl(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__add(X1,X2)) -> add(X1,X2)
       activate(n__s(X)) -> s(X)
       activate(n__dbl(X)) -> dbl(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
      Open