YES

Problem:
 min(x,0()) -> 0()
 min(0(),y) -> 0()
 min(s(x),s(y)) -> s(min(x,y))
 max(x,0()) -> x
 max(0(),y) -> y
 max(s(x),s(y)) -> s(max(x,y))
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 gcd(s(x),0()) -> s(x)
 gcd(0(),s(x)) -> s(x)
 gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

Proof:
 DP Processor:
  DPs:
   min#(s(x),s(y)) -> min#(x,y)
   max#(s(x),s(y)) -> max#(x,y)
   -#(s(x),s(y)) -> -#(x,y)
   gcd#(s(x),s(y)) -> min#(x,y)
   gcd#(s(x),s(y)) -> max#(x,y)
   gcd#(s(x),s(y)) -> -#(max(x,y),min(x,y))
   gcd#(s(x),s(y)) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)))
  TRS:
   min(x,0()) -> 0()
   min(0(),y) -> 0()
   min(s(x),s(y)) -> s(min(x,y))
   max(x,0()) -> x
   max(0(),y) -> y
   max(s(x),s(y)) -> s(max(x,y))
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   gcd(s(x),0()) -> s(x)
   gcd(0(),s(x)) -> s(x)
   gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [gcd#](x0, x1) = 4x0 + 2x1 + 1,
    
    [-#](x0, x1) = 5x1,
    
    [max#](x0, x1) = x1,
    
    [min#](x0, x1) = x0 + 2x1 + 3,
    
    [gcd](x0, x1) = 2x0 + x1 + 6,
    
    [-](x0, x1) = x0 + 2x1,
    
    [max](x0, x1) = x0 + 3x1,
    
    [s](x0) = 6x0 + 1,
    
    [min](x0, x1) = x0,
    
    [0] = 0
   orientation:
    min#(s(x),s(y)) = 6x + 12y + 6 >= x + 2y + 3 = min#(x,y)
    
    max#(s(x),s(y)) = 6y + 1 >= y = max#(x,y)
    
    -#(s(x),s(y)) = 30y + 5 >= 5y = -#(x,y)
    
    gcd#(s(x),s(y)) = 24x + 12y + 7 >= x + 2y + 3 = min#(x,y)
    
    gcd#(s(x),s(y)) = 24x + 12y + 7 >= y = max#(x,y)
    
    gcd#(s(x),s(y)) = 24x + 12y + 7 >= 5x = -#(max(x,y),min(x,y))
    
    gcd#(s(x),s(y)) = 24x + 12y + 7 >= 24x + 12y + 3 = gcd#(-(max(x,y),min(x,y)),s(min(x,y)))
    
    min(x,0()) = x >= 0 = 0()
    
    min(0(),y) = 0 >= 0 = 0()
    
    min(s(x),s(y)) = 6x + 1 >= 6x + 1 = s(min(x,y))
    
    max(x,0()) = x >= x = x
    
    max(0(),y) = 3y >= y = y
    
    max(s(x),s(y)) = 6x + 18y + 4 >= 6x + 18y + 1 = s(max(x,y))
    
    -(x,0()) = x >= x = x
    
    -(s(x),s(y)) = 6x + 12y + 3 >= x + 2y = -(x,y)
    
    gcd(s(x),0()) = 12x + 8 >= 6x + 1 = s(x)
    
    gcd(0(),s(x)) = 6x + 7 >= 6x + 1 = s(x)
    
    gcd(s(x),s(y)) = 12x + 6y + 9 >= 12x + 6y + 7 = gcd(-(max(x,y),min(x,y)),s(min(x,y)))
   problem:
    DPs:
     
    TRS:
     min(x,0()) -> 0()
     min(0(),y) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(x,0()) -> x
     max(0(),y) -> y
     max(s(x),s(y)) -> s(max(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     gcd(s(x),0()) -> s(x)
     gcd(0(),s(x)) -> s(x)
     gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))
   Qed