YES
Time: 2.566080
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}
 DP:
  DP:
   {             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}
  UR:
   {               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),
                      a(z, w) -> z,
                      a(z, w) -> w}
   EDG:
    {(mod#(s x, s y) -> le#(y, x), le#(s x, s y) -> le#(x, 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) -> mod#(minus(x, y), s y), mod#(s x, s y) -> le#(y, x))
     (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))
     (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) -> 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))
     (le#(s x, s y) -> le#(x, y), le#(s x, s y) -> le#(x, y))
     (minus#(s x, y) -> le#(s x, y), le#(s x, s y) -> le#(x, y))
     (minus#(s x, y) -> if_minus#(le(s x, y), s x, y), if_minus#(false(), s x, y) -> minus#(x, y))}
    STATUS:
     arrows: 0.812500
     SCCS (3):
      Scc:
       {           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)}
      Scc:
       {            minus#(s x, y) -> if_minus#(le(s x, y), s x, y),
        if_minus#(false(), s x, y) -> minus#(x, y)}
      Scc:
       {le#(s x, s y) -> le#(x, y)}
      
      SCC (2):
       Strict:
        {           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)}
       Weak:
       {               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}
       POLY:
        Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
        Interpretation:
         [if_minus](x0, x1, x2) = x0,
         
         [if_mod](x0, x1, x2) = x0 + 1,
         
         [le](x0, x1) = 0,
         
         [minus](x0, x1) = x0,
         
         [mod](x0, x1) = x0 + 1,
         
         [s](x0) = x0 + 1,
         
         [true] = 0,
         
         [0] = 0,
         
         [false] = 0,
         
         [if_mod#](x0, x1, x2) = x0 + 1,
         
         [mod#](x0, x1) = x0 + 1
        Strict:
         if_mod#(true(), s x, s y) -> mod#(minus(x, y), s y)
         2 + 0y + 1x >= 1 + 0y + 1x
         mod#(s x, s y) -> if_mod#(le(y, x), s x, s y)
         2 + 0y + 1x >= 2 + 0y + 1x
        Weak:
         if_mod(false(), s x, s y) -> s x
         2 + 0y + 1x >= 1 + 1x
         if_mod(true(), s x, s y) -> mod(minus(x, y), s y)
         2 + 0y + 1x >= 1 + 0y + 1x
         mod(s x, s y) -> if_mod(le(y, x), s x, s y)
         2 + 0y + 1x >= 2 + 0y + 1x
         mod(s x, 0()) -> 0()
         2 + 1x >= 0
         mod(0(), y) -> 0()
         1 + 0y >= 0
         if_minus(false(), s x, y) -> s minus(x, y)
         1 + 0y + 1x >= 1 + 0y + 1x
         if_minus(true(), s x, y) -> 0()
         1 + 0y + 1x >= 0
         minus(s x, y) -> if_minus(le(s x, y), s x, y)
         1 + 0y + 1x >= 1 + 0y + 1x
         minus(0(), y) -> 0()
         0 + 0y >= 0
         le(s x, s y) -> le(x, y)
         0 + 0y + 0x >= 0 + 0y + 0x
         le(s x, 0()) -> false()
         0 + 0x >= 0
         le(0(), y) -> true()
         0 + 0y >= 0
       SCCS (0):
        
        
     
     
     SCC (2):
      Strict:
       {            minus#(s x, y) -> if_minus#(le(s x, y), s x, y),
        if_minus#(false(), s x, y) -> minus#(x, y)}
      Weak:
      {               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}
      POLY:
       Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
       Interpretation:
        [if_minus](x0, x1, x2) = 0,
        
        [if_mod](x0, x1, x2) = x0 + 1,
        
        [le](x0, x1) = 0,
        
        [minus](x0, x1) = 0,
        
        [mod](x0, x1) = x0 + 1,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [0] = 1,
        
        [false] = 0,
        
        [if_minus#](x0, x1, x2) = x0,
        
        [minus#](x0, x1) = x0
       Strict:
        if_minus#(false(), s x, y) -> minus#(x, y)
        1 + 0y + 1x >= 0 + 0y + 1x
        minus#(s x, y) -> if_minus#(le(s x, y), s x, y)
        1 + 0y + 1x >= 1 + 0y + 1x
       Weak:
        if_mod(false(), s x, s y) -> s x
        2 + 0y + 1x >= 1 + 1x
        if_mod(true(), s x, s y) -> mod(minus(x, y), s y)
        2 + 0y + 1x >= 1 + 0y + 0x
        mod(s x, s y) -> if_mod(le(y, x), s x, s y)
        2 + 0y + 1x >= 2 + 0y + 1x
        mod(s x, 0()) -> 0()
        2 + 1x >= 1
        mod(0(), y) -> 0()
        2 + 0y >= 1
        if_minus(false(), s x, y) -> s minus(x, y)
        0 + 0y + 0x >= 1 + 0y + 0x
        if_minus(true(), s x, y) -> 0()
        0 + 0y + 0x >= 1
        minus(s x, y) -> if_minus(le(s x, y), s x, y)
        0 + 0y + 0x >= 0 + 0y + 0x
        minus(0(), y) -> 0()
        0 + 0y >= 1
        le(s x, s y) -> le(x, y)
        0 + 0y + 0x >= 0 + 0y + 0x
        le(s x, 0()) -> false()
        0 + 0x >= 0
        le(0(), y) -> true()
        0 + 0y >= 0
      SCCS (0):
       
       
     
     SCC (1):
      Strict:
       {le#(s x, s y) -> le#(x, y)}
      Weak:
      {               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}
      POLY:
       Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true
       Interpretation:
        [if_minus](x0, x1, x2) = x0,
        
        [if_mod](x0, x1, x2) = x0 + x1,
        
        [le](x0, x1) = x0 + 1,
        
        [minus](x0, x1) = x0 + 1,
        
        [mod](x0, x1) = x0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [0] = 1,
        
        [false] = 0,
        
        [le#](x0, x1) = x0 + 1
       Strict:
        le#(s x, s y) -> le#(x, y)
        2 + 0y + 1x >= 1 + 0y + 1x
       Weak:
        if_mod(false(), s x, s y) -> s x
        1 + 0y + 1x >= 1 + 1x
        if_mod(true(), s x, s y) -> mod(minus(x, y), s y)
        1 + 0y + 1x >= 1 + 1y + 0x
        mod(s x, s y) -> if_mod(le(y, x), s x, s y)
        1 + 1y + 0x >= 2 + 1y + 1x
        mod(s x, 0()) -> 0()
        1 + 0x >= 1
        mod(0(), y) -> 0()
        0 + 1y >= 1
        if_minus(false(), s x, y) -> s minus(x, y)
        0 + 0y + 0x >= 2 + 0y + 1x
        if_minus(true(), s x, y) -> 0()
        0 + 0y + 0x >= 1
        minus(s x, y) -> if_minus(le(s x, y), s x, y)
        2 + 0y + 1x >= 2 + 0y + 1x
        minus(0(), y) -> 0()
        2 + 0y >= 1
        le(s x, s y) -> le(x, y)
        2 + 0y + 1x >= 1 + 0y + 1x
        le(s x, 0()) -> false()
        2 + 1x >= 0
        le(0(), y) -> true()
        2 + 0y >= 0
      Qed