YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 pred(s(x)) -> x
 minus(x,0()) -> x
 minus(x,s(y)) -> pred(minus(x,y))
 mod(0(),y) -> 0()
 mod(s(x),0()) -> 0()
 mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
 if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
 if_mod(false(),s(x),s(y)) -> s(x)

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(x,s(y)) -> minus#(x,y)
   minus#(x,s(y)) -> pred#(minus(x,y))
   mod#(s(x),s(y)) -> le#(y,x)
   mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
   if_mod#(true(),s(x),s(y)) -> minus#(x,y)
   if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   pred(s(x)) -> x
   minus(x,0()) -> x
   minus(x,s(y)) -> pred(minus(x,y))
   mod(0(),y) -> 0()
   mod(s(x),0()) -> 0()
   mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
   if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
   if_mod(false(),s(x),s(y)) -> s(x)
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(x,s(y)) -> minus#(x,y)
    minus#(x,s(y)) -> pred#(minus(x,y))
    mod#(s(x),s(y)) -> le#(y,x)
    mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
    if_mod#(true(),s(x),s(y)) -> minus#(x,y)
    if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    pred(s(x)) -> x
    minus(x,0()) -> x
    minus(x,s(y)) -> pred(minus(x,y))
    mod(0(),y) -> 0()
    mod(s(x),0()) -> 0()
    mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
    if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
    if_mod(false(),s(x),s(y)) -> s(x)
   graph:
    if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) ->
    mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
    if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) ->
    mod#(s(x),s(y)) -> le#(y,x)
    if_mod#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(x,s(y)) -> pred#(minus(x,y))
    if_mod#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(x,s(y)) -> minus#(x,y)
    mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) ->
    if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
    mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) ->
    if_mod#(true(),s(x),s(y)) -> minus#(x,y)
    mod#(s(x),s(y)) -> le#(y,x) -> le#(s(x),s(y)) -> le#(x,y)
    minus#(x,s(y)) -> minus#(x,y) ->
    minus#(x,s(y)) -> pred#(minus(x,y))
    minus#(x,s(y)) -> minus#(x,y) -> minus#(x,s(y)) -> minus#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 10/49
    DPs:
     if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
     mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     pred(s(x)) -> x
     minus(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     mod(0(),y) -> 0()
     mod(s(x),0()) -> 0()
     mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
     if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
     if_mod(false(),s(x),s(y)) -> s(x)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [if_mod#](x0, x1, x2) = x0 + x1 + 2,
      
      [mod#](x0, x1) = 4x0 + 1,
      
      [if_mod](x0, x1, x2) = 2x1 + x2 + 1,
      
      [mod](x0, x1) = 4x0 + x1 + 1,
      
      [minus](x0, x1) = x0,
      
      [pred](x0) = x0,
      
      [false] = 0,
      
      [s](x0) = 4x0 + 1,
      
      [true] = 0,
      
      [le](x0, x1) = 7x1 + 1,
      
      [0] = 4
     orientation:
      if_mod#(true(),s(x),s(y)) = 4x + 3 >= 4x + 1 = mod#(minus(x,y),s(y))
      
      mod#(s(x),s(y)) = 16x + 5 >= 11x + 4 = if_mod#(le(y,x),s(x),s(y))
      
      le(0(),y) = 7y + 1 >= 0 = true()
      
      le(s(x),0()) = 29 >= 0 = false()
      
      le(s(x),s(y)) = 28y + 8 >= 7y + 1 = le(x,y)
      
      pred(s(x)) = 4x + 1 >= x = x
      
      minus(x,0()) = x >= x = x
      
      minus(x,s(y)) = x >= x = pred(minus(x,y))
      
      mod(0(),y) = y + 17 >= 4 = 0()
      
      mod(s(x),0()) = 16x + 9 >= 4 = 0()
      
      mod(s(x),s(y)) = 16x + 4y + 6 >= 8x + 4y + 4 = if_mod(le(y,x),s(x),s(y))
      
      if_mod(true(),s(x),s(y)) = 8x + 4y + 4 >= 4x + 4y + 2 = mod(minus(x,y),s(y))
      
      if_mod(false(),s(x),s(y)) = 8x + 4y + 4 >= 4x + 1 = s(x)
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       pred(s(x)) -> x
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
       if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
       if_mod(false(),s(x),s(y)) -> s(x)
     Qed
    
    DPs:
     minus#(x,s(y)) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     pred(s(x)) -> x
     minus(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     mod(0(),y) -> 0()
     mod(s(x),0()) -> 0()
     mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
     if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
     if_mod(false(),s(x),s(y)) -> s(x)
    KBO Processor:
     argument filtering:
      pi(0) = []
      pi(le) = 0
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(pred) = 0
      pi(minus) = 0
      pi(mod) = 0
      pi(if_mod) = 1
      pi(minus#) = 1
     weight function:
      w0 = 1
      w(minus#) = w(if_mod) = w(mod) = w(minus) = w(pred) = w(false) = w(
      s) = w(true) = w(le) = w(0) = 1
     precedence:
      minus# ~ minus ~ pred ~ s ~ 0 > if_mod ~ mod ~ false ~ true ~ le
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       pred(s(x)) -> x
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
       if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
       if_mod(false(),s(x),s(y)) -> s(x)
     Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     pred(s(x)) -> x
     minus(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     mod(0(),y) -> 0()
     mod(s(x),0()) -> 0()
     mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
     if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
     if_mod(false(),s(x),s(y)) -> s(x)
    KBO Processor:
     argument filtering:
      pi(0) = []
      pi(le) = [1]
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(pred) = 0
      pi(minus) = 0
      pi(mod) = 0
      pi(if_mod) = 1
      pi(le#) = 1
     weight function:
      w0 = 1
      w(le#) = w(mod) = w(minus) = w(pred) = w(false) = w(s) = w(true) = w(
      le) = w(0) = 1
      w(if_mod) = 0
     precedence:
      le# ~ if_mod ~ mod ~ minus ~ pred ~ false ~ s ~ true ~ le ~ 0
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       pred(s(x)) -> x
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
       if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
       if_mod(false(),s(x),s(y)) -> s(x)
     Qed