YES(?,O(n^1))

Problem:
 f(j(x,y),y) -> g(f(x,k(y)))
 f(x,h1(y,z)) -> h2(0(),x,h1(y,z))
 g(h2(x,y,h1(z,u))) -> h2(s(x),y,h1(z,u))
 h2(x,j(y,h1(z,u)),h1(z,u)) -> h2(s(x),y,h1(s(z),u))
 i(f(x,h(y))) -> y
 i(h2(s(x),y,h1(x,z))) -> z
 k(h(x)) -> h1(0(),x)
 k(h1(x,y)) -> h1(s(x),y)

Proof:
 RT Transformation Processor:
  strict:
   f(j(x,y),y) -> g(f(x,k(y)))
   f(x,h1(y,z)) -> h2(0(),x,h1(y,z))
   g(h2(x,y,h1(z,u))) -> h2(s(x),y,h1(z,u))
   h2(x,j(y,h1(z,u)),h1(z,u)) -> h2(s(x),y,h1(s(z),u))
   i(f(x,h(y))) -> y
   i(h2(s(x),y,h1(x,z))) -> z
   k(h(x)) -> h1(0(),x)
   k(h1(x,y)) -> h1(s(x),y)
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [i](x0) = x0 + 2,
    
    [h](x0) = x0 + 18,
    
    [s](x0) = x0 + 18,
    
    [h2](x0, x1, x2) = x0 + x1 + x2 + 4,
    
    [0] = 0,
    
    [h1](x0, x1) = x0 + x1,
    
    [g](x0) = x0 + 5,
    
    [k](x0) = x0 + 19,
    
    [f](x0, x1) = x0 + x1 + 8,
    
    [j](x0, x1) = x0 + x1 + 16
   orientation:
    f(j(x,y),y) = x + 2y + 24 >= x + y + 32 = g(f(x,k(y)))
    
    f(x,h1(y,z)) = x + y + z + 8 >= x + y + z + 4 = h2(0(),x,h1(y,z))
    
    g(h2(x,y,h1(z,u))) = u + x + y + z + 9 >= u + x + y + z + 22 = h2(s(x),y,h1(z,u))
    
    h2(x,j(y,h1(z,u)),h1(z,u)) = 2u + x + y + 2z + 20 >= u + x + y + z + 40 = h2(s(x),y,h1(s(z),u))
    
    i(f(x,h(y))) = x + y + 28 >= y = y
    
    i(h2(s(x),y,h1(x,z))) = 2x + y + z + 24 >= z = z
    
    k(h(x)) = x + 37 >= x = h1(0(),x)
    
    k(h1(x,y)) = x + y + 19 >= x + y + 18 = h1(s(x),y)
   problem:
    strict:
     f(j(x,y),y) -> g(f(x,k(y)))
     g(h2(x,y,h1(z,u))) -> h2(s(x),y,h1(z,u))
     h2(x,j(y,h1(z,u)),h1(z,u)) -> h2(s(x),y,h1(s(z),u))
    weak:
     f(x,h1(y,z)) -> h2(0(),x,h1(y,z))
     i(f(x,h(y))) -> y
     i(h2(s(x),y,h1(x,z))) -> z
     k(h(x)) -> h1(0(),x)
     k(h1(x,y)) -> h1(s(x),y)
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [i](x0) = x0,
     
     [h](x0) = x0 + 17,
     
     [s](x0) = x0 + 3,
     
     [h2](x0, x1, x2) = x0 + x1 + x2 + 18,
     
     [0] = 6,
     
     [h1](x0, x1) = x0 + x1 + 14,
     
     [g](x0) = x0,
     
     [k](x0) = x0 + 3,
     
     [f](x0, x1) = x0 + x1 + 24,
     
     [j](x0, x1) = x0 + x1 + 17
    orientation:
     f(j(x,y),y) = x + 2y + 41 >= x + y + 27 = g(f(x,k(y)))
     
     g(h2(x,y,h1(z,u))) = u + x + y + z + 32 >= u + x + y + z + 35 = h2(s(x),y,h1(z,u))
     
     h2(x,j(y,h1(z,u)),h1(z,u)) = 2u + x + y + 2z + 63 >= u + x + y + z + 38 = h2(s(x),y,h1(s(z),u))
     
     f(x,h1(y,z)) = x + y + z + 38 >= x + y + z + 38 = h2(0(),x,h1(y,z))
     
     i(f(x,h(y))) = x + y + 41 >= y = y
     
     i(h2(s(x),y,h1(x,z))) = 2x + y + z + 35 >= z = z
     
     k(h(x)) = x + 20 >= x + 20 = h1(0(),x)
     
     k(h1(x,y)) = x + y + 17 >= x + y + 17 = h1(s(x),y)
    problem:
     strict:
      g(h2(x,y,h1(z,u))) -> h2(s(x),y,h1(z,u))
     weak:
      f(j(x,y),y) -> g(f(x,k(y)))
      h2(x,j(y,h1(z,u)),h1(z,u)) -> h2(s(x),y,h1(s(z),u))
      f(x,h1(y,z)) -> h2(0(),x,h1(y,z))
      i(f(x,h(y))) -> y
      i(h2(s(x),y,h1(x,z))) -> z
      k(h(x)) -> h1(0(),x)
      k(h1(x,y)) -> h1(s(x),y)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [i](x0) = x0,
      
      [h](x0) = x0 + 24,
      
      [s](x0) = x0 + 1,
      
      [h2](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [0] = 7,
      
      [h1](x0, x1) = x0 + x1,
      
      [g](x0) = x0 + 14,
      
      [k](x0) = x0 + 1,
      
      [f](x0, x1) = x0 + x1 + 16,
      
      [j](x0, x1) = x0 + x1 + 16
     orientation:
      g(h2(x,y,h1(z,u))) = u + x + y + z + 15 >= u + x + y + z + 2 = h2(s(x),y,h1(z,u))
      
      f(j(x,y),y) = x + 2y + 32 >= x + y + 31 = g(f(x,k(y)))
      
      h2(x,j(y,h1(z,u)),h1(z,u)) = 2u + x + y + 2z + 17 >= u + x + y + z + 3 = h2(s(x),y,h1(s(z),u))
      
      f(x,h1(y,z)) = x + y + z + 16 >= x + y + z + 8 = h2(0(),x,h1(y,z))
      
      i(f(x,h(y))) = x + y + 40 >= y = y
      
      i(h2(s(x),y,h1(x,z))) = 2x + y + z + 2 >= z = z
      
      k(h(x)) = x + 25 >= x + 7 = h1(0(),x)
      
      k(h1(x,y)) = x + y + 1 >= x + y + 1 = h1(s(x),y)
     problem:
      strict:
       
      weak:
       g(h2(x,y,h1(z,u))) -> h2(s(x),y,h1(z,u))
       f(j(x,y),y) -> g(f(x,k(y)))
       h2(x,j(y,h1(z,u)),h1(z,u)) -> h2(s(x),y,h1(s(z),u))
       f(x,h1(y,z)) -> h2(0(),x,h1(y,z))
       i(f(x,h(y))) -> y
       i(h2(s(x),y,h1(x,z))) -> z
       k(h(x)) -> h1(0(),x)
       k(h1(x,y)) -> h1(s(x),y)
     Qed