YES

Problem:
 +(x,0()) -> x
 +(0(),x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 *(x,0()) -> 0()
 *(0(),x) -> 0()
 *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
 sum(nil()) -> 0()
 sum(cons(x,l)) -> +(x,sum(l))
 prod(nil()) -> s(0())
 prod(cons(x,l)) -> *(x,prod(l))

Proof:
 DP Processor:
  DPs:
   +#(s(x),s(y)) -> +#(x,y)
   *#(s(x),s(y)) -> +#(x,y)
   *#(s(x),s(y)) -> *#(x,y)
   *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
   sum#(cons(x,l)) -> sum#(l)
   sum#(cons(x,l)) -> +#(x,sum(l))
   prod#(cons(x,l)) -> prod#(l)
   prod#(cons(x,l)) -> *#(x,prod(l))
  TRS:
   +(x,0()) -> x
   +(0(),x) -> x
   +(s(x),s(y)) -> s(s(+(x,y)))
   *(x,0()) -> 0()
   *(0(),x) -> 0()
   *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
   sum(nil()) -> 0()
   sum(cons(x,l)) -> +(x,sum(l))
   prod(nil()) -> s(0())
   prod(cons(x,l)) -> *(x,prod(l))
  TDG Processor:
   DPs:
    +#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> *#(x,y)
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    sum#(cons(x,l)) -> sum#(l)
    sum#(cons(x,l)) -> +#(x,sum(l))
    prod#(cons(x,l)) -> prod#(l)
    prod#(cons(x,l)) -> *#(x,prod(l))
   TRS:
    +(x,0()) -> x
    +(0(),x) -> x
    +(s(x),s(y)) -> s(s(+(x,y)))
    *(x,0()) -> 0()
    *(0(),x) -> 0()
    *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
    sum(nil()) -> 0()
    sum(cons(x,l)) -> +(x,sum(l))
    prod(nil()) -> s(0())
    prod(cons(x,l)) -> *(x,prod(l))
   graph:
    prod#(cons(x,l)) -> prod#(l) -> prod#(cons(x,l)) -> *#(x,prod(l))
    prod#(cons(x,l)) -> prod#(l) -> prod#(cons(x,l)) -> prod#(l)
    prod#(cons(x,l)) -> *#(x,prod(l)) ->
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(s(x),s(y)) -> *#(x,y)
    prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(s(x),s(y)) -> +#(x,y)
    sum#(cons(x,l)) -> sum#(l) -> sum#(cons(x,l)) -> +#(x,sum(l))
    sum#(cons(x,l)) -> sum#(l) -> sum#(cons(x,l)) -> sum#(l)
    sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> *#(x,y) -> *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
    *#(s(x),s(y)) -> *#(x,y) -> *#(s(x),s(y)) -> *#(x,y)
    *#(s(x),s(y)) -> *#(x,y) -> *#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> +#(*(x,y),+(x,y)) -> +#(s(x),s(y)) -> +#(x,y)
    *#(s(x),s(y)) -> +#(x,y) -> +#(s(x),s(y)) -> +#(x,y)
    +#(s(x),s(y)) -> +#(x,y) -> +#(s(x),s(y)) -> +#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 4
    #arcs: 14/64
    DPs:
     sum#(cons(x,l)) -> sum#(l)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [sum#](x0) = x0,
      
      [prod](x0) = x0 + 1,
      
      [cons](x0, x1) = 4x0 + 2x1 + 3,
      
      [sum](x0) = 1x0 + 2,
      
      [nil] = 2,
      
      [*](x0, x1) = 2,
      
      [s](x0) = 0,
      
      [+](x0, x1) = 1x0 + 1x1 + 2,
      
      [0] = 0
     orientation:
      sum#(cons(x,l)) = 2l + 4x + 3 >= l = sum#(l)
      
      +(x,0()) = 1x + 2 >= x = x
      
      +(0(),x) = 1x + 2 >= x = x
      
      +(s(x),s(y)) = 2 >= 0 = s(s(+(x,y)))
      
      *(x,0()) = 2 >= 0 = 0()
      
      *(0(),x) = 2 >= 0 = 0()
      
      *(s(x),s(y)) = 2 >= 0 = s(+(*(x,y),+(x,y)))
      
      sum(nil()) = 3 >= 0 = 0()
      
      sum(cons(x,l)) = 3l + 5x + 4 >= 2l + 1x + 3 = +(x,sum(l))
      
      prod(nil()) = 2 >= 0 = s(0())
      
      prod(cons(x,l)) = 2l + 4x + 3 >= 2 = *(x,prod(l))
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
     Qed
    
    DPs:
     prod#(cons(x,l)) -> prod#(l)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [prod#](x0) = x0,
      
      [prod](x0) = x0 + 1,
      
      [cons](x0, x1) = 4x0 + 2x1 + 3,
      
      [sum](x0) = 1x0 + 2,
      
      [nil] = 2,
      
      [*](x0, x1) = 2,
      
      [s](x0) = 0,
      
      [+](x0, x1) = 1x0 + 1x1 + 2,
      
      [0] = 0
     orientation:
      prod#(cons(x,l)) = 2l + 4x + 3 >= l = prod#(l)
      
      +(x,0()) = 1x + 2 >= x = x
      
      +(0(),x) = 1x + 2 >= x = x
      
      +(s(x),s(y)) = 2 >= 0 = s(s(+(x,y)))
      
      *(x,0()) = 2 >= 0 = 0()
      
      *(0(),x) = 2 >= 0 = 0()
      
      *(s(x),s(y)) = 2 >= 0 = s(+(*(x,y),+(x,y)))
      
      sum(nil()) = 3 >= 0 = 0()
      
      sum(cons(x,l)) = 3l + 5x + 4 >= 2l + 1x + 3 = +(x,sum(l))
      
      prod(nil()) = 2 >= 0 = s(0())
      
      prod(cons(x,l)) = 2l + 4x + 3 >= 2 = *(x,prod(l))
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
     Qed
    
    DPs:
     *#(s(x),s(y)) -> *#(x,y)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
    LPO Processor:
     argument filtering:
      pi(0) = []
      pi(+) = [0,1]
      pi(s) = [0]
      pi(*) = [0,1]
      pi(nil) = []
      pi(sum) = [0]
      pi(cons) = [0,1]
      pi(prod) = 0
      pi(*#) = 0
     precedence:
      cons > * > sum ~ nil ~ + > *# ~ prod ~ s ~ 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
     Qed
    
    DPs:
     +#(s(x),s(y)) -> +#(x,y)
    TRS:
     +(x,0()) -> x
     +(0(),x) -> x
     +(s(x),s(y)) -> s(s(+(x,y)))
     *(x,0()) -> 0()
     *(0(),x) -> 0()
     *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
     sum(nil()) -> 0()
     sum(cons(x,l)) -> +(x,sum(l))
     prod(nil()) -> s(0())
     prod(cons(x,l)) -> *(x,prod(l))
    LPO Processor:
     argument filtering:
      pi(0) = []
      pi(+) = [0,1]
      pi(s) = [0]
      pi(*) = [0,1]
      pi(nil) = []
      pi(sum) = [0]
      pi(cons) = [0,1]
      pi(prod) = 0
      pi(+#) = 0
     precedence:
      cons > * > sum ~ nil ~ + > +# ~ prod ~ s ~ 0
     problem:
      DPs:
       
      TRS:
       +(x,0()) -> x
       +(0(),x) -> x
       +(s(x),s(y)) -> s(s(+(x,y)))
       *(x,0()) -> 0()
       *(0(),x) -> 0()
       *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
       sum(nil()) -> 0()
       sum(cons(x,l)) -> +(x,sum(l))
       prod(nil()) -> s(0())
       prod(cons(x,l)) -> *(x,prod(l))
     Qed