YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,0()) -> x
 minus(s(x),s(y)) -> 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),s(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(x,0()) -> x
   minus(s(x),s(y)) -> 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),s(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(x,0()) -> x
    minus(s(x),s(y)) -> 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),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#(s(x),s(y)) -> minus#(x,y) -> minus#(s(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: 8/36
    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(x,0()) -> x
     minus(s(x),s(y)) -> 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)
    Usable Rule Processor:
     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:
      minus(x,0()) -> x
      minus(s(x),s(y)) -> minus(x,y)
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
      interpretation:
       [if_mod#](x0, x1, x2) = x1 + 0,
       
       [mod#](x0, x1) = 2x0 + 0,
       
       [minus](x0, x1) = x0 + 0,
       
       [false] = 0,
       
       [s](x0) = 2x0 + 2,
       
       [true] = 0,
       
       [le](x0, x1) = 2x0 + 2x1 + 4,
       
       [0] = 0
      orientation:
       if_mod#(true(),s(x),s(y)) = 2x + 2 >= 2x + 2 = mod#(minus(x,y),s(y))
       
       mod#(s(x),s(y)) = 4x + 4 >= 2x + 2 = if_mod#(le(y,x),s(x),s(y))
       
       minus(x,0()) = x + 0 >= x = x
       
       minus(s(x),s(y)) = 2x + 2 >= x + 0 = minus(x,y)
       
       le(0(),y) = 2y + 4 >= 0 = true()
       
       le(s(x),0()) = 4x + 4 >= 0 = false()
       
       le(s(x),s(y)) = 4x + 4y + 4 >= 2x + 2y + 4 = le(x,y)
      problem:
       DPs:
        if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
       TRS:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        le(0(),y) -> true()
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
      Restore Modifier:
       DPs:
        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(x,0()) -> x
        minus(s(x),s(y)) -> 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)
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 2/1
        
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,0()) -> x
     minus(s(x),s(y)) -> 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)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> 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(x,0()) -> x
     minus(s(x),s(y)) -> 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)
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> 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