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))
 gcd(0(),y) -> y
 gcd(s(x),0()) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),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)
   gcd#(s(x),s(y)) -> le#(y,x)
   gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
   if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
   if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
   if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
   if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
  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))
   gcd(0(),y) -> y
   gcd(s(x),0()) -> s(x)
   gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
   if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
   if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [if_gcd#](x0, x1, x2) = 4x1 + 2x2 + 4,
    
    [gcd#](x0, x1) = 4x0 + 2x1 + 5,
    
    [if_minus#](x0, x1, x2) = x0 + 4x1 + 3x2,
    
    [minus#](x0, x1) = 4x0 + 3x1 + 3,
    
    [le#](x0, x1) = 2x1 + 4,
    
    [if_gcd](x0, x1, x2) = 3x0 + x1 + x2 + 3,
    
    [gcd](x0, x1) = 2x0 + x1 + 5,
    
    [if_minus](x0, x1, x2) = 2x1 + 1,
    
    [minus](x0, x1) = 2x0 + 1,
    
    [false] = 0,
    
    [s](x0) = 4x0 + 4,
    
    [true] = 0,
    
    [le](x0, x1) = 2,
    
    [0] = 0
   orientation:
    le#(s(x),s(y)) = 8y + 12 >= 2y + 4 = le#(x,y)
    
    minus#(s(x),y) = 16x + 3y + 19 >= 2y + 4 = le#(s(x),y)
    
    minus#(s(x),y) = 16x + 3y + 19 >= 16x + 3y + 18 = if_minus#(le(s(x),y),s(x),y)
    
    if_minus#(false(),s(x),y) = 16x + 3y + 16 >= 4x + 3y + 3 = minus#(x,y)
    
    gcd#(s(x),s(y)) = 16x + 8y + 29 >= 2x + 4 = le#(y,x)
    
    gcd#(s(x),s(y)) = 16x + 8y + 29 >= 16x + 8y + 28 = if_gcd#(le(y,x),s(x),s(y))
    
    if_gcd#(true(),s(x),s(y)) = 16x + 8y + 28 >= 4x + 3y + 3 = minus#(x,y)
    
    if_gcd#(true(),s(x),s(y)) = 16x + 8y + 28 >= 8x + 8y + 17 = gcd#(minus(x,y),s(y))
    
    if_gcd#(false(),s(x),s(y)) = 16x + 8y + 28 >= 3x + 4y + 3 = minus#(y,x)
    
    if_gcd#(false(),s(x),s(y)) = 16x + 8y + 28 >= 8x + 8y + 17 = gcd#(minus(y,x),s(x))
    
    le(0(),y) = 2 >= 0 = true()
    
    le(s(x),0()) = 2 >= 0 = false()
    
    le(s(x),s(y)) = 2 >= 2 = le(x,y)
    
    minus(0(),y) = 1 >= 0 = 0()
    
    minus(s(x),y) = 8x + 9 >= 8x + 9 = if_minus(le(s(x),y),s(x),y)
    
    if_minus(true(),s(x),y) = 8x + 9 >= 0 = 0()
    
    if_minus(false(),s(x),y) = 8x + 9 >= 8x + 8 = s(minus(x,y))
    
    gcd(0(),y) = y + 5 >= y = y
    
    gcd(s(x),0()) = 8x + 13 >= 4x + 4 = s(x)
    
    gcd(s(x),s(y)) = 8x + 4y + 17 >= 4x + 4y + 17 = if_gcd(le(y,x),s(x),s(y))
    
    if_gcd(true(),s(x),s(y)) = 4x + 4y + 11 >= 4x + 4y + 11 = gcd(minus(x,y),s(y))
    
    if_gcd(false(),s(x),s(y)) = 4x + 4y + 11 >= 4x + 4y + 11 = gcd(minus(y,x),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))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
   Qed