YES

Problem:
 gcd(x,0()) -> x
 gcd(0(),y) -> y
 gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))

Proof:
 DP Processor:
  DPs:
   gcd#(s(x),s(y)) -> gcd#(-(x,y),s(y))
   gcd#(s(x),s(y)) -> gcd#(s(x),-(y,x))
  TRS:
   gcd(x,0()) -> x
   gcd(0(),y) -> y
   gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
  Usable Rule Processor:
   DPs:
    gcd#(s(x),s(y)) -> gcd#(-(x,y),s(y))
    gcd#(s(x),s(y)) -> gcd#(s(x),-(y,x))
   TRS:
    f7(x,y) -> x
    f7(x,y) -> y
   Restore Modifier:
    DPs:
     gcd#(s(x),s(y)) -> gcd#(-(x,y),s(y))
     gcd#(s(x),s(y)) -> gcd#(s(x),-(y,x))
    TRS:
     gcd(x,0()) -> x
     gcd(0(),y) -> y
     gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 4/4
     DPs:
      gcd#(s(x),s(y)) -> gcd#(-(x,y),s(y))
      gcd#(s(x),s(y)) -> gcd#(s(x),-(y,x))
     TRS:
      gcd(x,0()) -> x
      gcd(0(),y) -> y
      gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [gcd#](x0, x1) = x1,
       
       [if](x0, x1, x2) = 1,
       
       [-](x0, x1) = 0,
       
       [<](x0, x1) = 0,
       
       [s](x0) = x0 + 1,
       
       [gcd](x0, x1) = x0 + x1 + 1,
       
       [0] = 0
      orientation:
       gcd#(s(x),s(y)) = y + 1 >= y + 1 = gcd#(-(x,y),s(y))
       
       gcd#(s(x),s(y)) = y + 1 >= 0 = gcd#(s(x),-(y,x))
       
       gcd(x,0()) = x + 1 >= x = x
       
       gcd(0(),y) = y + 1 >= y = y
       
       gcd(s(x),s(y)) = x + y + 3 >= 1 = if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
      problem:
       DPs:
        gcd#(s(x),s(y)) -> gcd#(-(x,y),s(y))
       TRS:
        gcd(x,0()) -> x
        gcd(0(),y) -> y
        gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [gcd#](x0, x1) = x0,
        
        [if](x0, x1, x2) = x0 + 1,
        
        [-](x0, x1) = 0,
        
        [<](x0, x1) = 1,
        
        [s](x0) = x0 + 1,
        
        [gcd](x0, x1) = x0 + x1,
        
        [0] = 0
       orientation:
        gcd#(s(x),s(y)) = x + 1 >= 0 = gcd#(-(x,y),s(y))
        
        gcd(x,0()) = x >= x = x
        
        gcd(0(),y) = y >= y = y
        
        gcd(s(x),s(y)) = x + y + 2 >= 2 = if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
       problem:
        DPs:
         
        TRS:
         gcd(x,0()) -> x
         gcd(0(),y) -> y
         gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))
       Qed