YES

Problem:
 not(x) -> xor(x,true())
 or(x,y) -> xor(and(x,y),xor(x,y))
 implies(x,y) -> xor(and(x,y),xor(x,true()))
 and(x,true()) -> x
 and(x,false()) -> false()
 and(x,x) -> x
 xor(x,false()) -> x
 xor(x,x) -> false()
 and(xor(x,y),z) -> xor(and(x,z),and(y,z))

Proof:
 DP Processor:
  DPs:
   not#(x) -> xor#(x,true())
   or#(x,y) -> xor#(x,y)
   or#(x,y) -> and#(x,y)
   or#(x,y) -> xor#(and(x,y),xor(x,y))
   implies#(x,y) -> xor#(x,true())
   implies#(x,y) -> and#(x,y)
   implies#(x,y) -> xor#(and(x,y),xor(x,true()))
   and#(xor(x,y),z) -> and#(y,z)
   and#(xor(x,y),z) -> and#(x,z)
   and#(xor(x,y),z) -> xor#(and(x,z),and(y,z))
  TRS:
   not(x) -> xor(x,true())
   or(x,y) -> xor(and(x,y),xor(x,y))
   implies(x,y) -> xor(and(x,y),xor(x,true()))
   and(x,true()) -> x
   and(x,false()) -> false()
   and(x,x) -> x
   xor(x,false()) -> x
   xor(x,x) -> false()
   and(xor(x,y),z) -> xor(and(x,z),and(y,z))
  Usable Rule Processor:
   DPs:
    not#(x) -> xor#(x,true())
    or#(x,y) -> xor#(x,y)
    or#(x,y) -> and#(x,y)
    or#(x,y) -> xor#(and(x,y),xor(x,y))
    implies#(x,y) -> xor#(x,true())
    implies#(x,y) -> and#(x,y)
    implies#(x,y) -> xor#(and(x,y),xor(x,true()))
    and#(xor(x,y),z) -> and#(y,z)
    and#(xor(x,y),z) -> and#(x,z)
    and#(xor(x,y),z) -> xor#(and(x,z),and(y,z))
   TRS:
    xor(x,false()) -> x
    xor(x,x) -> false()
    and(x,true()) -> x
    and(x,false()) -> false()
    and(x,x) -> x
    and(xor(x,y),z) -> xor(and(x,z),and(y,z))
   Matrix Interpretation Processor: dim=1
    
    usable rules:
     
    interpretation:
     [implies#](x0, x1) = 2x0 + x1 + 7,
     
     [and#](x0, x1) = 2x0 + x1 + 6,
     
     [or#](x0, x1) = 3x0 + 4x1 + 7,
     
     [xor#](x0, x1) = 0,
     
     [not#](x0) = 4x0 + 4,
     
     [false] = 2,
     
     [and](x0, x1) = x0 + 4x1,
     
     [xor](x0, x1) = x0 + 3x1 + 4,
     
     [true] = 0
    orientation:
     not#(x) = 4x + 4 >= 0 = xor#(x,true())
     
     or#(x,y) = 3x + 4y + 7 >= 0 = xor#(x,y)
     
     or#(x,y) = 3x + 4y + 7 >= 2x + y + 6 = and#(x,y)
     
     or#(x,y) = 3x + 4y + 7 >= 0 = xor#(and(x,y),xor(x,y))
     
     implies#(x,y) = 2x + y + 7 >= 0 = xor#(x,true())
     
     implies#(x,y) = 2x + y + 7 >= 2x + y + 6 = and#(x,y)
     
     implies#(x,y) = 2x + y + 7 >= 0 = xor#(and(x,y),xor(x,true()))
     
     and#(xor(x,y),z) = 2x + 6y + z + 14 >= 2y + z + 6 = and#(y,z)
     
     and#(xor(x,y),z) = 2x + 6y + z + 14 >= 2x + z + 6 = and#(x,z)
     
     and#(xor(x,y),z) = 2x + 6y + z + 14 >= 0 = xor#(and(x,z),and(y,z))
     
     xor(x,false()) = x + 10 >= x = x
     
     xor(x,x) = 4x + 4 >= 2 = false()
     
     and(x,true()) = x >= x = x
     
     and(x,false()) = x + 8 >= 2 = false()
     
     and(x,x) = 5x >= x = x
     
     and(xor(x,y),z) = x + 3y + 4z + 4 >= x + 3y + 16z + 4 = xor(and(x,z),and(y,z))
    problem:
     DPs:
      
     TRS:
      xor(x,false()) -> x
      xor(x,x) -> false()
      and(x,true()) -> x
      and(x,false()) -> false()
      and(x,x) -> x
      and(xor(x,y),z) -> xor(and(x,z),and(y,z))
    Qed