YES

Problem:
 d(0()) -> 0()
 d(s(x)) -> s(s(d(x)))
 e(0()) -> s(0())
 e(r(x)) -> d(e(x))

Proof:
 DP Processor:
  DPs:
   d#(s(x)) -> d#(x)
   e#(r(x)) -> e#(x)
   e#(r(x)) -> d#(e(x))
  TRS:
   d(0()) -> 0()
   d(s(x)) -> s(s(d(x)))
   e(0()) -> s(0())
   e(r(x)) -> d(e(x))
  TDG Processor:
   DPs:
    d#(s(x)) -> d#(x)
    e#(r(x)) -> e#(x)
    e#(r(x)) -> d#(e(x))
   TRS:
    d(0()) -> 0()
    d(s(x)) -> s(s(d(x)))
    e(0()) -> s(0())
    e(r(x)) -> d(e(x))
   graph:
    e#(r(x)) -> e#(x) -> e#(r(x)) -> d#(e(x))
    e#(r(x)) -> e#(x) -> e#(r(x)) -> e#(x)
    e#(r(x)) -> d#(e(x)) -> d#(s(x)) -> d#(x)
    d#(s(x)) -> d#(x) -> d#(s(x)) -> d#(x)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 4/9
    DPs:
     e#(r(x)) -> e#(x)
    TRS:
     d(0()) -> 0()
     d(s(x)) -> s(s(d(x)))
     e(0()) -> s(0())
     e(r(x)) -> d(e(x))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [e#](x0) = -8x0 + 2,
      
      [r](x0) = 8x0 + 12,
      
      [e](x0) = -7x0 + 4,
      
      [s](x0) = 0,
      
      [d](x0) = x0 + 2,
      
      [0] = 0
     orientation:
      e#(r(x)) = x + 4 >= -8x + 2 = e#(x)
      
      d(0()) = 2 >= 0 = 0()
      
      d(s(x)) = 2 >= 0 = s(s(d(x)))
      
      e(0()) = 4 >= 0 = s(0())
      
      e(r(x)) = 1x + 5 >= -7x + 4 = d(e(x))
     problem:
      DPs:
       
      TRS:
       d(0()) -> 0()
       d(s(x)) -> s(s(d(x)))
       e(0()) -> s(0())
       e(r(x)) -> d(e(x))
     Qed
    
    DPs:
     d#(s(x)) -> d#(x)
    TRS:
     d(0()) -> 0()
     d(s(x)) -> s(s(d(x)))
     e(0()) -> s(0())
     e(r(x)) -> d(e(x))
    LPO Processor:
     argument filtering:
      pi(0) = []
      pi(d) = [0]
      pi(s) = [0]
      pi(e) = [0]
      pi(r) = [0]
      pi(d#) = 0
     precedence:
      e > d > d# ~ r ~ s ~ 0
     problem:
      DPs:
       
      TRS:
       d(0()) -> 0()
       d(s(x)) -> s(s(d(x)))
       e(0()) -> s(0())
       e(r(x)) -> d(e(x))
     Qed