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

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),z) -> min#(x,y)
   gcd#(s(x),s(y),z) -> max#(x,y)
   gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
   gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
   gcd#(x,s(y),s(z)) -> min#(y,z)
   gcd#(x,s(y),s(z)) -> max#(y,z)
   gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
   gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
   gcd#(s(x),y,s(z)) -> min#(x,z)
   gcd#(s(x),y,s(z)) -> max#(x,z)
   gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
   gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
  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),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z)
   gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z)))
   gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z)))
   gcd(x,0(),0()) -> x
   gcd(0(),y,0()) -> y
   gcd(0(),0(),z) -> z
  Usable Rule 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),z) -> min#(x,y)
    gcd#(s(x),s(y),z) -> max#(x,y)
    gcd#(s(x),s(y),z) -> -#(max(x,y),min(x,y))
    gcd#(s(x),s(y),z) -> gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
    gcd#(x,s(y),s(z)) -> min#(y,z)
    gcd#(x,s(y),s(z)) -> max#(y,z)
    gcd#(x,s(y),s(z)) -> -#(max(y,z),min(y,z))
    gcd#(x,s(y),s(z)) -> gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
    gcd#(s(x),y,s(z)) -> min#(x,z)
    gcd#(s(x),y,s(z)) -> max#(x,z)
    gcd#(s(x),y,s(z)) -> -#(max(x,z),min(x,z))
    gcd#(s(x),y,s(z)) -> gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
   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)
   Matrix Interpretation Processor: dim=1
    
    usable rules:
     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)
    interpretation:
     [gcd#](x0, x1, x2) = 1/2x0 + 1/2x1 + 1/2x2 + 7/2,
     
     [-#](x0, x1) = 2x1 + 1,
     
     [max#](x0, x1) = x0 + 2,
     
     [min#](x0, x1) = x0 + 1/2x1,
     
     [-](x0, x1) = x0,
     
     [max](x0, x1) = x0 + x1,
     
     [s](x0) = 2x0 + 1,
     
     [min](x0, x1) = 1/2x0 + 1/2x1,
     
     [0] = 0
    orientation:
     min#(s(x),s(y)) = 2x + y + 3/2 >= x + 1/2y = min#(x,y)
     
     max#(s(x),s(y)) = 2x + 3 >= x + 2 = max#(x,y)
     
     -#(s(x),s(y)) = 4y + 3 >= 2y + 1 = -#(x,y)
     
     gcd#(s(x),s(y),z) = x + y + 1/2z + 9/2 >= x + 1/2y = min#(x,y)
     
     gcd#(s(x),s(y),z) = x + y + 1/2z + 9/2 >= x + 2 = max#(x,y)
     
     gcd#(s(x),s(y),z) = x + y + 1/2z + 9/2 >= x + y + 1 = -#(max(x,y),min(x,y))
     
     gcd#(s(x),s(y),z) = x + y + 1/2z + 9/2 >= x + y + 1/2z + 4 = gcd#(-(max(x,y),min(x,y)),s(min(x,y)),z)
     
     gcd#(x,s(y),s(z)) = 1/2x + y + z + 9/2 >= y + 1/2z = min#(y,z)
     
     gcd#(x,s(y),s(z)) = 1/2x + y + z + 9/2 >= y + 2 = max#(y,z)
     
     gcd#(x,s(y),s(z)) = 1/2x + y + z + 9/2 >= y + z + 1 = -#(max(y,z),min(y,z))
     
     gcd#(x,s(y),s(z)) = 1/2x + y + z + 9/2 >= 1/2x + y + z + 4 = gcd#(x,-(max(y,z),min(y,z)),s(min(y,z)))
     
     gcd#(s(x),y,s(z)) = x + 1/2y + z + 9/2 >= x + 1/2z = min#(x,z)
     
     gcd#(s(x),y,s(z)) = x + 1/2y + z + 9/2 >= x + 2 = max#(x,z)
     
     gcd#(s(x),y,s(z)) = x + 1/2y + z + 9/2 >= x + z + 1 = -#(max(x,z),min(x,z))
     
     gcd#(s(x),y,s(z)) = x + 1/2y + z + 9/2 >= x + 1/2y + z + 4 = gcd#(-(max(x,z),min(x,z)),y,s(min(x,z)))
     
     min(x,0()) = 1/2x >= 0 = 0()
     
     min(0(),y) = 1/2y >= 0 = 0()
     
     min(s(x),s(y)) = x + y + 1 >= x + y + 1 = s(min(x,y))
     
     max(x,0()) = x >= x = x
     
     max(0(),y) = y >= y = y
     
     max(s(x),s(y)) = 2x + 2y + 2 >= 2x + 2y + 1 = s(max(x,y))
     
     -(x,0()) = x >= x = x
     
     -(s(x),s(y)) = 2x + 1 >= x = -(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)
    Qed