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)
    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(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))
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      le(0(),y) -> true()
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       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))
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       le(0(),y) -> true()
      interpretation:
       [if_mod#](x0, x1, x2) = -4x0 + 2x1,
       
       [mod#](x0, x1) = 3x0 + 1,
       
       [if_minus](x0, x1, x2) = -8x0 + x1 + -4,
       
       [minus](x0, x1) = x0 + -3,
       
       [false] = 3,
       
       [s](x0) = 3x0 + 0,
       
       [true] = 3,
       
       [le](x0, x1) = 4,
       
       [0] = 0
      orientation:
       if_mod#(true(),s(x),s(y)) = 5x + 2 >= 3x + 1 = mod#(minus(x,y),s(y))
       
       mod#(s(x),s(y)) = 6x + 3 >= 5x + 2 = if_mod#(le(y,x),s(x),s(y))
       
       minus(0(),y) = 0 >= 0 = 0()
       
       minus(s(x),y) = 3x + 0 >= 3x + 0 = if_minus(le(s(x),y),s(x),y)
       
       if_minus(true(),s(x),y) = 3x + 0 >= 0 = 0()
       
       if_minus(false(),s(x),y) = 3x + 0 >= 3x + 0 = s(minus(x,y))
       
       le(s(x),0()) = 4 >= 3 = false()
       
       le(s(x),s(y)) = 4 >= 4 = le(x,y)
       
       le(0(),y) = 4 >= 3 = true()
      problem:
       DPs:
        
       TRS:
        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))
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
        le(0(),y) -> true()
      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)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(if_minus#) = 1
     problem:
      DPs:
       minus#(s(x),y) -> if_minus#(le(s(x),y),s(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)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    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)
    Subterm Criterion Processor:
     simple projection:
      pi(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