/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

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
     usable rules:
      terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
      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
     interpretation:
      [first#](x0, x1) = x0 + x1 + 0,
      
      [add#](x0, x1) = x0 + 1x1 + 0,
      
      [activate#](x0) = x0 + 0,
      
      [dbl#](x0) = x0 + 0,
      
      [sqr#](x0) = x0 + 0,
      
      [terms#](x0) = x0 + 0,
      
      [n__first](x0, x1) = 2x0 + 1x1 + 4,
      
      [nil] = 0,
      
      [first](x0, x1) = 2x0 + 1x1 + 4,
      
      [add](x0, x1) = x0 + 1x1 + 3,
      
      [n__s](x0) = x0,
      
      [n__dbl](x0) = 1x0 + 2,
      
      [n__add](x0, x1) = x0 + 1x1 + 3,
      
      [dbl](x0) = 1x0 + 2,
      
      [activate](x0) = x0 + 0,
      
      [0] = 3,
      
      [cons](x0, x1) = x1,
      
      [n__terms](x0) = 1x0,
      
      [s](x0) = x0,
      
      [recip](x0) = 2x0 + 0,
      
      [sqr](x0) = 2x0 + 2,
      
      [terms](x0) = 1x0
     orientation:
      first#(s(X),cons(Y,Z)) = X + Z + 0 >= Z + 0 = activate#(Z)
      
      activate#(n__terms(X)) = 1X + 0 >= X + 0 = terms#(X)
      
      terms#(N) = N + 0 >= N + 0 = sqr#(N)
      
      sqr#(s(X)) = X + 0 >= X + 0 = dbl#(activate(X))
      
      dbl#(s(X)) = X + 0 >= X + 0 = activate#(X)
      
      activate#(n__add(X1,X2)) = X1 + 1X2 + 3 >= X1 + 1X2 + 0 = add#(X1,X2)
      
      add#(s(X),Y) = X + 1Y + 0 >= X + 0 = activate#(X)
      
      activate#(n__dbl(X)) = 1X + 2 >= X + 0 = dbl#(X)
      
      activate#(n__first(X1,X2)) = 2X1 + 1X2 + 4 >= X1 + X2 + 0 = first#(X1,X2)
      
      first#(s(X),cons(Y,Z)) = X + Z + 0 >= X + 0 = activate#(X)
      
      sqr#(s(X)) = X + 0 >= X + 0 = activate#(X)
      
      sqr#(s(X)) = X + 0 >= X + 0 = sqr#(activate(X))
      
      terms(N) = 1N >= 1N = cons(recip(sqr(N)),n__terms(s(N)))
      
      sqr(0()) = 5 >= 3 = 0()
      
      sqr(s(X)) = 2X + 2 >= 2X + 3 = s(n__add(sqr(activate(X)),dbl(activate(X))))
      
      dbl(0()) = 4 >= 3 = 0()
      
      dbl(s(X)) = 1X + 2 >= 1X + 2 = s(n__s(n__dbl(activate(X))))
      
      add(0(),X) = 1X + 3 >= X = X
      
      add(s(X),Y) = X + 1Y + 3 >= X + 1Y + 3 = s(n__add(activate(X),Y))
      
      first(0(),X) = 1X + 5 >= 0 = nil()
      
      first(s(X),cons(Y,Z)) = 2X + 1Z + 4 >= 2X + 1Z + 4 = cons(Y,n__first(activate(X),activate(Z)))
      
      terms(X) = 1X >= 1X = n__terms(X)
      
      add(X1,X2) = X1 + 1X2 + 3 >= X1 + 1X2 + 3 = n__add(X1,X2)
      
      s(X) = X >= X = n__s(X)
      
      dbl(X) = 1X + 2 >= 1X + 2 = n__dbl(X)
      
      first(X1,X2) = 2X1 + 1X2 + 4 >= 2X1 + 1X2 + 4 = n__first(X1,X2)
      
      activate(n__terms(X)) = 1X + 0 >= 1X = terms(X)
      
      activate(n__add(X1,X2)) = X1 + 1X2 + 3 >= X1 + 1X2 + 3 = add(X1,X2)
      
      activate(n__s(X)) = X + 0 >= X = s(X)
      
      activate(n__dbl(X)) = 1X + 2 >= 1X + 2 = dbl(X)
      
      activate(n__first(X1,X2)) = 2X1 + 1X2 + 4 >= 2X1 + 1X2 + 4 = 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)
       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)
       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
     Restore Modifier:
      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)
       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
      SCC Processor:
       #sccs: 1
       #rules: 8
       #arcs: 32/100
       DPs:
        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)
        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
        usable rules:
         terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
         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
        interpretation:
         [add#](x0, x1) = 5x0 + x1,
         
         [activate#](x0) = x0,
         
         [dbl#](x0) = x0 + 0,
         
         [sqr#](x0) = x0,
         
         [terms#](x0) = x0,
         
         [n__first](x0, x1) = x0 + x1,
         
         [nil] = 0,
         
         [first](x0, x1) = x0 + x1,
         
         [add](x0, x1) = 5x0 + x1 + 7,
         
         [n__s](x0) = x0,
         
         [n__dbl](x0) = x0,
         
         [n__add](x0, x1) = 5x0 + x1 + 7,
         
         [dbl](x0) = x0,
         
         [activate](x0) = x0 + 0,
         
         [0] = 0,
         
         [cons](x0, x1) = x1,
         
         [n__terms](x0) = x0 + 4,
         
         [s](x0) = x0 + 0,
         
         [recip](x0) = x0 + 0,
         
         [sqr](x0) = x0 + 2,
         
         [terms](x0) = x0 + 4
        orientation:
         activate#(n__terms(X)) = X + 4 >= X = terms#(X)
         
         terms#(N) = N >= N = sqr#(N)
         
         sqr#(s(X)) = X + 0 >= X + 0 = dbl#(activate(X))
         
         dbl#(s(X)) = X + 0 >= X = activate#(X)
         
         activate#(n__add(X1,X2)) = 5X1 + X2 + 7 >= 5X1 + X2 = add#(X1,X2)
         
         add#(s(X),Y) = 5X + Y + 5 >= X = activate#(X)
         
         sqr#(s(X)) = X + 0 >= X = activate#(X)
         
         sqr#(s(X)) = X + 0 >= X + 0 = sqr#(activate(X))
         
         terms(N) = N + 4 >= N + 4 = cons(recip(sqr(N)),n__terms(s(N)))
         
         sqr(0()) = 2 >= 0 = 0()
         
         sqr(s(X)) = X + 2 >= 5X + 7 = 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 + 7 >= X = X
         
         add(s(X),Y) = 5X + Y + 7 >= 5X + Y + 7 = s(n__add(activate(X),Y))
         
         first(0(),X) = X + 0 >= 0 = nil()
         
         first(s(X),cons(Y,Z)) = X + Z + 0 >= X + Z + 0 = cons(Y,n__first(activate(X),activate(Z)))
         
         terms(X) = X + 4 >= X + 4 = n__terms(X)
         
         add(X1,X2) = 5X1 + X2 + 7 >= 5X1 + X2 + 7 = n__add(X1,X2)
         
         s(X) = X + 0 >= X = n__s(X)
         
         dbl(X) = X >= X = n__dbl(X)
         
         first(X1,X2) = X1 + X2 >= X1 + X2 = n__first(X1,X2)
         
         activate(n__terms(X)) = X + 4 >= X + 4 = terms(X)
         
         activate(n__add(X1,X2)) = 5X1 + X2 + 7 >= 5X1 + X2 + 7 = add(X1,X2)
         
         activate(n__s(X)) = X + 0 >= X + 0 = s(X)
         
         activate(n__dbl(X)) = X + 0 >= X = dbl(X)
         
         activate(n__first(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = first(X1,X2)
         
         activate(X) = X + 0 >= X = X
        problem:
         DPs:
          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)
          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
        Restore Modifier:
         DPs:
          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)
          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
         SCC Processor:
          #sccs: 1
          #rules: 6
          #arcs: 15/49
          DPs:
           activate#(n__terms(X)) -> terms#(X)
           terms#(N) -> sqr#(N)
           sqr#(s(X)) -> dbl#(activate(X))
           dbl#(s(X)) -> 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
           usable rules:
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            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
           interpretation:
            [activate#](x0) = x0,
            
            [dbl#](x0) = x0,
            
            [sqr#](x0) = 1x0,
            
            [terms#](x0) = 1x0,
            
            [n__first](x0, x1) = x0 + 2x1,
            
            [nil] = 0,
            
            [first](x0, x1) = x0 + 2x1,
            
            [add](x0, x1) = 2x0 + 2x1 + 0,
            
            [n__s](x0) = x0,
            
            [n__dbl](x0) = 0,
            
            [n__add](x0, x1) = 2x0 + 2x1 + 0,
            
            [dbl](x0) = 0,
            
            [activate](x0) = x0,
            
            [0] = 0,
            
            [cons](x0, x1) = x0 + x1 + 0,
            
            [n__terms](x0) = 1x0 + 2,
            
            [s](x0) = x0,
            
            [recip](x0) = 0,
            
            [sqr](x0) = 0,
            
            [terms](x0) = 1x0 + 2
           orientation:
            activate#(n__terms(X)) = 1X + 2 >= 1X = terms#(X)
            
            terms#(N) = 1N >= 1N = sqr#(N)
            
            sqr#(s(X)) = 1X >= X = dbl#(activate(X))
            
            dbl#(s(X)) = X >= X = activate#(X)
            
            sqr#(s(X)) = 1X >= X = activate#(X)
            
            sqr#(s(X)) = 1X >= 1X = sqr#(activate(X))
            
            terms(N) = 1N + 2 >= 1N + 2 = cons(recip(sqr(N)),n__terms(s(N)))
            
            sqr(0()) = 0 >= 0 = 0()
            
            sqr(s(X)) = 0 >= 2 = s(n__add(sqr(activate(X)),dbl(activate(X))))
            
            dbl(0()) = 0 >= 0 = 0()
            
            dbl(s(X)) = 0 >= 0 = s(n__s(n__dbl(activate(X))))
            
            add(0(),X) = 2X + 2 >= X = X
            
            add(s(X),Y) = 2X + 2Y + 0 >= 2X + 2Y + 0 = s(n__add(activate(X),Y))
            
            first(0(),X) = 2X + 0 >= 0 = nil()
            
            first(s(X),cons(Y,Z)) = X + 2Y + 2Z + 2 >= X + Y + 2Z + 0 = cons(Y,n__first(activate(X),activate(Z)))
            
            terms(X) = 1X + 2 >= 1X + 2 = n__terms(X)
            
            add(X1,X2) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = n__add(X1,X2)
            
            s(X) = X >= X = n__s(X)
            
            dbl(X) = 0 >= 0 = n__dbl(X)
            
            first(X1,X2) = X1 + 2X2 >= X1 + 2X2 = n__first(X1,X2)
            
            activate(n__terms(X)) = 1X + 2 >= 1X + 2 = terms(X)
            
            activate(n__add(X1,X2)) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = add(X1,X2)
            
            activate(n__s(X)) = X >= X = s(X)
            
            activate(n__dbl(X)) = 0 >= 0 = dbl(X)
            
            activate(n__first(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = first(X1,X2)
            
            activate(X) = X >= X = X
           problem:
            DPs:
             activate#(n__terms(X)) -> terms#(X)
             terms#(N) -> sqr#(N)
             dbl#(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
           Restore Modifier:
            DPs:
             activate#(n__terms(X)) -> terms#(X)
             terms#(N) -> sqr#(N)
             dbl#(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
            SCC Processor:
             #sccs: 1
             #rules: 1
             #arcs: 10/16
             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
             Matrix Interpretation Processor: dim=1
              
              usable rules:
               terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
               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
              interpretation:
               [sqr#](x0) = x0,
               
               [n__first](x0, x1) = 5x0 + 3,
               
               [nil] = 3,
               
               [first](x0, x1) = 5x0 + 3,
               
               [add](x0, x1) = 4x0 + 2x1,
               
               [n__s](x0) = x0 + 1,
               
               [n__dbl](x0) = 2x0,
               
               [n__add](x0, x1) = 4x0 + 2x1,
               
               [dbl](x0) = 2x0,
               
               [activate](x0) = x0,
               
               [0] = 0,
               
               [cons](x0, x1) = 0,
               
               [n__terms](x0) = 0,
               
               [s](x0) = x0 + 1,
               
               [recip](x0) = 0,
               
               [sqr](x0) = 4x0,
               
               [terms](x0) = 0
              orientation:
               sqr#(s(X)) = X + 1 >= X = sqr#(activate(X))
               
               terms(N) = 0 >= 0 = cons(recip(sqr(N)),n__terms(s(N)))
               
               sqr(0()) = 0 >= 0 = 0()
               
               sqr(s(X)) = 4X + 4 >= 20X + 1 = s(n__add(sqr(activate(X)),dbl(activate(X))))
               
               dbl(0()) = 0 >= 0 = 0()
               
               dbl(s(X)) = 2X + 2 >= 2X + 2 = s(n__s(n__dbl(activate(X))))
               
               add(0(),X) = 2X >= X = X
               
               add(s(X),Y) = 4X + 2Y + 4 >= 4X + 2Y + 1 = s(n__add(activate(X),Y))
               
               first(0(),X) = 3 >= 3 = nil()
               
               first(s(X),cons(Y,Z)) = 5X + 8 >= 0 = cons(Y,n__first(activate(X),activate(Z)))
               
               terms(X) = 0 >= 0 = n__terms(X)
               
               add(X1,X2) = 4X1 + 2X2 >= 4X1 + 2X2 = n__add(X1,X2)
               
               s(X) = X + 1 >= X + 1 = n__s(X)
               
               dbl(X) = 2X >= 2X = n__dbl(X)
               
               first(X1,X2) = 5X1 + 3 >= 5X1 + 3 = n__first(X1,X2)
               
               activate(n__terms(X)) = 0 >= 0 = terms(X)
               
               activate(n__add(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = add(X1,X2)
               
               activate(n__s(X)) = X + 1 >= X + 1 = s(X)
               
               activate(n__dbl(X)) = 2X >= 2X = dbl(X)
               
               activate(n__first(X1,X2)) = 5X1 + 3 >= 5X1 + 3 = first(X1,X2)
               
               activate(X) = X >= X = X
              problem:
               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
              Qed