YES

Problem:
 lcs(e(),y) -> 0()
 lcs(x,e()) -> 0()
 lcs(a(x),a(y)) -> s(lcs(x,y))
 lcs(b(x),b(y)) -> s(lcs(x,y))
 lcs(a(x),b(y)) -> max(lcs(x,b(y)),lcs(a(x),y))
 lcs(b(x),a(y)) -> max(lcs(x,a(y)),lcs(b(x),y))
 max(x,0()) -> 0()
 max(0(),y) -> 0()
 max(s(x),s(y)) -> max(x,y)

Proof:
 DP Processor:
  DPs:
   lcs#(a(x),a(y)) -> lcs#(x,y)
   lcs#(b(x),b(y)) -> lcs#(x,y)
   lcs#(a(x),b(y)) -> lcs#(a(x),y)
   lcs#(a(x),b(y)) -> lcs#(x,b(y))
   lcs#(a(x),b(y)) -> max#(lcs(x,b(y)),lcs(a(x),y))
   lcs#(b(x),a(y)) -> lcs#(b(x),y)
   lcs#(b(x),a(y)) -> lcs#(x,a(y))
   lcs#(b(x),a(y)) -> max#(lcs(x,a(y)),lcs(b(x),y))
   max#(s(x),s(y)) -> max#(x,y)
  TRS:
   lcs(e(),y) -> 0()
   lcs(x,e()) -> 0()
   lcs(a(x),a(y)) -> s(lcs(x,y))
   lcs(b(x),b(y)) -> s(lcs(x,y))
   lcs(a(x),b(y)) -> max(lcs(x,b(y)),lcs(a(x),y))
   lcs(b(x),a(y)) -> max(lcs(x,a(y)),lcs(b(x),y))
   max(x,0()) -> 0()
   max(0(),y) -> 0()
   max(s(x),s(y)) -> max(x,y)
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [max#](x0, x1) = 2x1,
    
    [lcs#](x0, x1) = 3x0 + x1 + 2,
    
    [max](x0, x1) = 7,
    
    [b](x0) = 6x0 + 2,
    
    [s](x0) = x0 + 1,
    
    [a](x0) = 4x0 + 4,
    
    [0] = 0,
    
    [lcs](x0, x1) = 2x1 + 3,
    
    [e] = 0
   orientation:
    lcs#(a(x),a(y)) = 12x + 4y + 18 >= 3x + y + 2 = lcs#(x,y)
    
    lcs#(b(x),b(y)) = 18x + 6y + 10 >= 3x + y + 2 = lcs#(x,y)
    
    lcs#(a(x),b(y)) = 12x + 6y + 16 >= 12x + y + 14 = lcs#(a(x),y)
    
    lcs#(a(x),b(y)) = 12x + 6y + 16 >= 3x + 6y + 4 = lcs#(x,b(y))
    
    lcs#(a(x),b(y)) = 12x + 6y + 16 >= 4y + 6 = max#(lcs(x,b(y)),lcs(a(x),y))
    
    lcs#(b(x),a(y)) = 18x + 4y + 12 >= 18x + y + 8 = lcs#(b(x),y)
    
    lcs#(b(x),a(y)) = 18x + 4y + 12 >= 3x + 4y + 6 = lcs#(x,a(y))
    
    lcs#(b(x),a(y)) = 18x + 4y + 12 >= 4y + 6 = max#(lcs(x,a(y)),lcs(b(x),y))
    
    max#(s(x),s(y)) = 2y + 2 >= 2y = max#(x,y)
    
    lcs(e(),y) = 2y + 3 >= 0 = 0()
    
    lcs(x,e()) = 3 >= 0 = 0()
    
    lcs(a(x),a(y)) = 8y + 11 >= 2y + 4 = s(lcs(x,y))
    
    lcs(b(x),b(y)) = 12y + 7 >= 2y + 4 = s(lcs(x,y))
    
    lcs(a(x),b(y)) = 12y + 7 >= 7 = max(lcs(x,b(y)),lcs(a(x),y))
    
    lcs(b(x),a(y)) = 8y + 11 >= 7 = max(lcs(x,a(y)),lcs(b(x),y))
    
    max(x,0()) = 7 >= 0 = 0()
    
    max(0(),y) = 7 >= 0 = 0()
    
    max(s(x),s(y)) = 7 >= 7 = max(x,y)
   problem:
    DPs:
     
    TRS:
     lcs(e(),y) -> 0()
     lcs(x,e()) -> 0()
     lcs(a(x),a(y)) -> s(lcs(x,y))
     lcs(b(x),b(y)) -> s(lcs(x,y))
     lcs(a(x),b(y)) -> max(lcs(x,b(y)),lcs(a(x),y))
     lcs(b(x),a(y)) -> max(lcs(x,a(y)),lcs(b(x),y))
     max(x,0()) -> 0()
     max(0(),y) -> 0()
     max(s(x),s(y)) -> max(x,y)
   Qed