YES
Time: 0.054045
TRS:
 {                 +(0(), y) -> y,
                   +(s x, y) -> s +(x, y),
               ++(nil(), ys) -> ys,
            ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
  sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
             sum :(x, nil()) -> :(x, nil()),
          sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                   -(x, 0()) -> x,
                 -(0(), s y) -> 0(),
                 -(s x, s y) -> -(x, y),
              quot(0(), s y) -> 0(),
              quot(s x, s y) -> s quot(-(x, y), s y),
                length nil() -> 0(),
             length :(x, xs) -> s length xs,
                 hd :(x, xs) -> x,
                      avg xs -> quot(hd sum xs, length xs)}
 DP:
  DP:
   {                 +#(s x, y) -> +#(x, y),
              ++#(:(x, xs), ys) -> ++#(xs, ys),
    sum# ++(xs, :(x, :(y, ys))) -> ++#(xs, sum :(x, :(y, ys))),
    sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))),
    sum# ++(xs, :(x, :(y, ys))) -> sum# :(x, :(y, ys)),
            sum# :(x, :(y, xs)) -> +#(x, y),
            sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs),
                   -#(s x, s y) -> -#(x, y),
                quot#(s x, s y) -> -#(x, y),
                quot#(s x, s y) -> quot#(-(x, y), s y),
               length# :(x, xs) -> length# xs,
                        avg# xs -> sum# xs,
                        avg# xs -> quot#(hd sum xs, length xs),
                        avg# xs -> length# xs,
                        avg# xs -> hd# sum xs}
  TRS:
  {                 +(0(), y) -> y,
                    +(s x, y) -> s +(x, y),
                ++(nil(), ys) -> ys,
             ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
   sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
              sum :(x, nil()) -> :(x, nil()),
           sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                    -(x, 0()) -> x,
                  -(0(), s y) -> 0(),
                  -(s x, s y) -> -(x, y),
               quot(0(), s y) -> 0(),
               quot(s x, s y) -> s quot(-(x, y), s y),
                 length nil() -> 0(),
              length :(x, xs) -> s length xs,
                  hd :(x, xs) -> x,
                       avg xs -> quot(hd sum xs, length xs)}
  EDG:
   {(quot#(s x, s y) -> quot#(-(x, y), s y), quot#(s x, s y) -> quot#(-(x, y), s y))
    (quot#(s x, s y) -> quot#(-(x, y), s y), quot#(s x, s y) -> -#(x, y))
    (+#(s x, y) -> +#(x, y), +#(s x, y) -> +#(x, y))
    (-#(s x, s y) -> -#(x, y), -#(s x, s y) -> -#(x, y))
    (length# :(x, xs) -> length# xs, length# :(x, xs) -> length# xs)
    (avg# xs -> length# xs, length# :(x, xs) -> length# xs)
    (sum# ++(xs, :(x, :(y, ys))) -> sum# :(x, :(y, ys)), sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs))
    (sum# ++(xs, :(x, :(y, ys))) -> sum# :(x, :(y, ys)), sum# :(x, :(y, xs)) -> +#(x, y))
    (sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs), sum# :(x, :(y, xs)) -> +#(x, y))
    (sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs), sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs))
    (sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))), sum# ++(xs, :(x, :(y, ys))) -> ++#(xs, sum :(x, :(y, ys))))
    (sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))), sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))))
    (sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))), sum# ++(xs, :(x, :(y, ys))) -> sum# :(x, :(y, ys)))
    (sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))), sum# :(x, :(y, xs)) -> +#(x, y))
    (sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))), sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs))
    (avg# xs -> sum# xs, sum# ++(xs, :(x, :(y, ys))) -> ++#(xs, sum :(x, :(y, ys))))
    (avg# xs -> sum# xs, sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys))))
    (avg# xs -> sum# xs, sum# ++(xs, :(x, :(y, ys))) -> sum# :(x, :(y, ys)))
    (avg# xs -> sum# xs, sum# :(x, :(y, xs)) -> +#(x, y))
    (avg# xs -> sum# xs, sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs))
    (quot#(s x, s y) -> -#(x, y), -#(s x, s y) -> -#(x, y))
    (sum# :(x, :(y, xs)) -> +#(x, y), +#(s x, y) -> +#(x, y))
    (++#(:(x, xs), ys) -> ++#(xs, ys), ++#(:(x, xs), ys) -> ++#(xs, ys))
    (avg# xs -> quot#(hd sum xs, length xs), quot#(s x, s y) -> -#(x, y))
    (avg# xs -> quot#(hd sum xs, length xs), quot#(s x, s y) -> quot#(-(x, y), s y))
    (sum# ++(xs, :(x, :(y, ys))) -> ++#(xs, sum :(x, :(y, ys))), ++#(:(x, xs), ys) -> ++#(xs, ys))}
   SCCS (7):
    Scc:
     {length# :(x, xs) -> length# xs}
    Scc:
     {quot#(s x, s y) -> quot#(-(x, y), s y)}
    Scc:
     {-#(s x, s y) -> -#(x, y)}
    Scc:
     {sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs)}
    Scc:
     {sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys)))}
    Scc:
     {++#(:(x, xs), ys) -> ++#(xs, ys)}
    Scc:
     {+#(s x, y) -> +#(x, y)}
    SCC (1):
     Strict:
      {length# :(x, xs) -> length# xs}
     Weak:
     {                 +(0(), y) -> y,
                       +(s x, y) -> s +(x, y),
                   ++(nil(), ys) -> ys,
                ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
      sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
                 sum :(x, nil()) -> :(x, nil()),
              sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                       -(x, 0()) -> x,
                     -(0(), s y) -> 0(),
                     -(s x, s y) -> -(x, y),
                  quot(0(), s y) -> 0(),
                  quot(s x, s y) -> s quot(-(x, y), s y),
                    length nil() -> 0(),
                 length :(x, xs) -> s length xs,
                     hd :(x, xs) -> x,
                          avg xs -> quot(hd sum xs, length xs)}
     SPSC:
      Simple Projection:
       pi(length#) = 0
      Strict:
       {}
      Qed
    SCC (1):
     Strict:
      {quot#(s x, s y) -> quot#(-(x, y), s y)}
     Weak:
     {                 +(0(), y) -> y,
                       +(s x, y) -> s +(x, y),
                   ++(nil(), ys) -> ys,
                ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
      sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
                 sum :(x, nil()) -> :(x, nil()),
              sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                       -(x, 0()) -> x,
                     -(0(), s y) -> 0(),
                     -(s x, s y) -> -(x, y),
                  quot(0(), s y) -> 0(),
                  quot(s x, s y) -> s quot(-(x, y), s y),
                    length nil() -> 0(),
                 length :(x, xs) -> s length xs,
                     hd :(x, xs) -> x,
                          avg xs -> quot(hd sum xs, length xs)}
     POLY:
      Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
      Interpretation:
       [+](x0, x1) = x0 + 1,
       
       [++](x0, x1) = x0 + 1,
       
       [:](x0, x1) = x0 + 1,
       
       [-](x0, x1) = x0,
       
       [quot](x0, x1) = x0 + 1,
       
       [s](x0) = x0 + 1,
       
       [sum](x0) = 0,
       
       [length](x0) = x0 + 1,
       
       [hd](x0) = 0,
       
       [avg](x0) = 0,
       
       [0] = 1,
       
       [nil] = 1,
       
       [quot#](x0, x1) = x0 + 1
      Strict:
       quot#(s x, s y) -> quot#(-(x, y), s y)
       2 + 0y + 1x >= 1 + 0y + 1x
      Weak:
       
     Qed
   SCC (1):
    Strict:
     {-#(s x, s y) -> -#(x, y)}
    Weak:
    {                 +(0(), y) -> y,
                      +(s x, y) -> s +(x, y),
                  ++(nil(), ys) -> ys,
               ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
     sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
                sum :(x, nil()) -> :(x, nil()),
             sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                      -(x, 0()) -> x,
                    -(0(), s y) -> 0(),
                    -(s x, s y) -> -(x, y),
                 quot(0(), s y) -> 0(),
                 quot(s x, s y) -> s quot(-(x, y), s y),
                   length nil() -> 0(),
                length :(x, xs) -> s length xs,
                    hd :(x, xs) -> x,
                         avg xs -> quot(hd sum xs, length xs)}
    SPSC:
     Simple Projection:
      pi(-#) = 1
     Strict:
      {}
     Qed
   SCC (1):
    Strict:
     {sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs)}
    Weak:
    {                 +(0(), y) -> y,
                      +(s x, y) -> s +(x, y),
                  ++(nil(), ys) -> ys,
               ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
     sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
                sum :(x, nil()) -> :(x, nil()),
             sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                      -(x, 0()) -> x,
                    -(0(), s y) -> 0(),
                    -(s x, s y) -> -(x, y),
                 quot(0(), s y) -> 0(),
                 quot(s x, s y) -> s quot(-(x, y), s y),
                   length nil() -> 0(),
                length :(x, xs) -> s length xs,
                    hd :(x, xs) -> x,
                         avg xs -> quot(hd sum xs, length xs)}
    POLY:
     Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
     Interpretation:
      [+](x0, x1) = x0 + 1,
      
      [++](x0, x1) = x0 + 1,
      
      [:](x0, x1) = x0 + 1,
      
      [-](x0, x1) = x0 + 1,
      
      [quot](x0, x1) = x0 + 1,
      
      [s](x0) = x0 + 1,
      
      [sum](x0) = 0,
      
      [length](x0) = 0,
      
      [hd](x0) = 0,
      
      [avg](x0) = 0,
      
      [0] = 1,
      
      [nil] = 1,
      
      [sum#](x0) = x0
     Strict:
      sum# :(x, :(y, xs)) -> sum# :(+(x, y), xs)
      2 + 0y + 0x + 1xs >= 1 + 0y + 0x + 1xs
     Weak:
      
    Qed
   SCC (1):
    Strict:
     {sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys)))}
    Weak:
    {                 +(0(), y) -> y,
                      +(s x, y) -> s +(x, y),
                  ++(nil(), ys) -> ys,
               ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
     sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
                sum :(x, nil()) -> :(x, nil()),
             sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                      -(x, 0()) -> x,
                    -(0(), s y) -> 0(),
                    -(s x, s y) -> -(x, y),
                 quot(0(), s y) -> 0(),
                 quot(s x, s y) -> s quot(-(x, y), s y),
                   length nil() -> 0(),
                length :(x, xs) -> s length xs,
                    hd :(x, xs) -> x,
                         avg xs -> quot(hd sum xs, length xs)}
    POLY:
     Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
     Interpretation:
      [+](x0, x1) = x0 + 1,
      
      [++](x0, x1) = x0 + x1 + 1,
      
      [:](x0, x1) = x0 + 1,
      
      [-](x0, x1) = x0 + 1,
      
      [quot](x0, x1) = x0 + 1,
      
      [s](x0) = x0 + 1,
      
      [sum](x0) = 1,
      
      [length](x0) = 0,
      
      [hd](x0) = 0,
      
      [avg](x0) = 0,
      
      [0] = 1,
      
      [nil] = 0,
      
      [sum#](x0) = x0
     Strict:
      sum# ++(xs, :(x, :(y, ys))) -> sum# ++(xs, sum :(x, :(y, ys)))
      3 + 0y + 0x + 1ys + 1xs >= 2 + 0y + 0x + 0ys + 1xs
     Weak:
      
    Qed
  SCC (1):
   Strict:
    {++#(:(x, xs), ys) -> ++#(xs, ys)}
   Weak:
   {                 +(0(), y) -> y,
                     +(s x, y) -> s +(x, y),
                 ++(nil(), ys) -> ys,
              ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
    sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
               sum :(x, nil()) -> :(x, nil()),
            sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                     -(x, 0()) -> x,
                   -(0(), s y) -> 0(),
                   -(s x, s y) -> -(x, y),
                quot(0(), s y) -> 0(),
                quot(s x, s y) -> s quot(-(x, y), s y),
                  length nil() -> 0(),
               length :(x, xs) -> s length xs,
                   hd :(x, xs) -> x,
                        avg xs -> quot(hd sum xs, length xs)}
   SPSC:
    Simple Projection:
     pi(++#) = 0
    Strict:
     {}
    Qed
  SCC (1):
   Strict:
    {+#(s x, y) -> +#(x, y)}
   Weak:
   {                 +(0(), y) -> y,
                     +(s x, y) -> s +(x, y),
                 ++(nil(), ys) -> ys,
              ++(:(x, xs), ys) -> :(x, ++(xs, ys)),
    sum ++(xs, :(x, :(y, ys))) -> sum ++(xs, sum :(x, :(y, ys))),
               sum :(x, nil()) -> :(x, nil()),
            sum :(x, :(y, xs)) -> sum :(+(x, y), xs),
                     -(x, 0()) -> x,
                   -(0(), s y) -> 0(),
                   -(s x, s y) -> -(x, y),
                quot(0(), s y) -> 0(),
                quot(s x, s y) -> s quot(-(x, y), s y),
                  length nil() -> 0(),
               length :(x, xs) -> s length xs,
                   hd :(x, xs) -> x,
                        avg xs -> quot(hd sum xs, length xs)}
   SPSC:
    Simple Projection:
     pi(+#) = 0
    Strict:
     {}
    Qed