YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(0(),y) -> 0()
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
 if_minus(true(),s(x),y) -> 0()
 if_minus(false(),s(x),y) -> s(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#(s(x),y) -> le#(s(x),y)
   minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
   if_minus#(false(),s(x),y) -> 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)
   minus(0(),y) -> 0()
   minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
   if_minus(true(),s(x),y) -> 0()
   if_minus(false(),s(x),y) -> s(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#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> 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)
    minus(0(),y) -> 0()
    minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
    if_minus(true(),s(x),y) -> 0()
    if_minus(false(),s(x),y) -> s(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#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_mod#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(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)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) ->
    if_minus#(false(),s(x),y) -> minus#(x,y)
    minus#(s(x),y) -> le#(s(x),y) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 5
    #arcs: 12/64
    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)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(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) = 7/2x1 + 2x2 + 3/2,
      
      [mod#](x0, x1) = 7/2x0 + 2x1 + 2,
      
      [if_mod](x0, x1, x2) = 3/2x1,
      
      [mod](x0, x1) = 3/2x0,
      
      [if_minus](x0, x1, x2) = x1,
      
      [minus](x0, x1) = x0,
      
      [false] = 0,
      
      [s](x0) = x0 + 1,
      
      [true] = 0,
      
      [le](x0, x1) = 0,
      
      [0] = 0
     orientation:
      if_mod#(true(),s(x),s(y)) = 7/2x + 2y + 7 >= 7/2x + 2y + 4 = mod#(minus(x,y),s(y))
      
      mod#(s(x),s(y)) = 7/2x + 2y + 15/2 >= 7/2x + 2y + 7 = if_mod#(le(y,x),s(x),s(y))
      
      le(0(),y) = 0 >= 0 = true()
      
      le(s(x),0()) = 0 >= 0 = false()
      
      le(s(x),s(y)) = 0 >= 0 = le(x,y)
      
      minus(0(),y) = 0 >= 0 = 0()
      
      minus(s(x),y) = x + 1 >= x + 1 = if_minus(le(s(x),y),s(x),y)
      
      if_minus(true(),s(x),y) = x + 1 >= 0 = 0()
      
      if_minus(false(),s(x),y) = x + 1 >= x + 1 = s(minus(x,y))
      
      mod(0(),y) = 0 >= 0 = 0()
      
      mod(s(x),0()) = 3/2x + 3/2 >= 0 = 0()
      
      mod(s(x),s(y)) = 3/2x + 3/2 >= 3/2x + 3/2 = if_mod(le(y,x),s(x),s(y))
      
      if_mod(true(),s(x),s(y)) = 3/2x + 3/2 >= 3/2x = mod(minus(x,y),s(y))
      
      if_mod(false(),s(x),s(y)) = 3/2x + 3/2 >= x + 1 = s(x)
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(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#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
     if_minus#(false(),s(x),y) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(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_minus#](x0, x1, x2) = 2x1,
      
      [minus#](x0, x1) = 3x0,
      
      [if_mod](x0, x1, x2) = x1,
      
      [mod](x0, x1) = x0,
      
      [if_minus](x0, x1, x2) = x1,
      
      [minus](x0, x1) = x0,
      
      [false] = 0,
      
      [s](x0) = 2x0 + 1/2,
      
      [true] = 0,
      
      [le](x0, x1) = 0,
      
      [0] = 0
     orientation:
      minus#(s(x),y) = 6x + 3/2 >= 4x + 1 = if_minus#(le(s(x),y),s(x),y)
      
      if_minus#(false(),s(x),y) = 4x + 1 >= 3x = minus#(x,y)
      
      le(0(),y) = 0 >= 0 = true()
      
      le(s(x),0()) = 0 >= 0 = false()
      
      le(s(x),s(y)) = 0 >= 0 = le(x,y)
      
      minus(0(),y) = 0 >= 0 = 0()
      
      minus(s(x),y) = 2x + 1/2 >= 2x + 1/2 = if_minus(le(s(x),y),s(x),y)
      
      if_minus(true(),s(x),y) = 2x + 1/2 >= 0 = 0()
      
      if_minus(false(),s(x),y) = 2x + 1/2 >= 2x + 1/2 = s(minus(x,y))
      
      mod(0(),y) = 0 >= 0 = 0()
      
      mod(s(x),0()) = 2x + 1/2 >= 0 = 0()
      
      mod(s(x),s(y)) = 2x + 1/2 >= 2x + 1/2 = if_mod(le(y,x),s(x),s(y))
      
      if_mod(true(),s(x),s(y)) = 2x + 1/2 >= x = mod(minus(x,y),s(y))
      
      if_mod(false(),s(x),s(y)) = 2x + 1/2 >= 2x + 1/2 = s(x)
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(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)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(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(minus) = 0
      pi(if_minus) = 1
      pi(mod) = 0
      pi(if_mod) = 1
      pi(le#) = 1
     weight function:
      w0 = 1
      w(le#) = w(if_mod) = w(mod) = w(if_minus) = w(minus) = w(false) = w(
      s) = w(true) = w(le) = w(0) = 1
     precedence:
      le# ~ if_mod ~ mod ~ if_minus ~ minus ~ 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)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(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