YES

Problem:
 ba0(0()) -> 0()
 ba1(0()) -> 1()
 S(0()) -> 1()
 S(1()) -> ba0(1())
 S(ba0(x)) -> ba1(x)
 S(ba1(x)) -> ba0(S(x))
 plus(x,0()) -> x
 plus(0(),x) -> x
 plus(x,1()) -> S(x)
 plus(1(),x) -> S(x)
 plus(ba0(x),ba0(y)) -> plus(ba0(plus(x,y)),0())
 plus(ba0(x),ba1(y)) -> plus(ba0(plus(x,y)),1())
 plus(ba1(x),ba0(y)) -> plus(ba1(plus(x,y)),0())
 plus(ba1(x),ba1(y)) -> plus(ba1(plus(x,y)),1())
 mult(x,0()) -> 0()
 mult(x,1()) -> x
 mult(x,ba0(y)) -> plus(ba0(mult(x,y)),mult(x,0()))
 mult(x,ba1(y)) -> plus(ba0(mult(x,y)),mult(x,1()))

Proof:
 Church Rosser Transformation Processor (kb):
  ba0(0()) -> 0()
  ba1(0()) -> 1()
  S(0()) -> 1()
  S(1()) -> ba0(1())
  S(ba0(x)) -> ba1(x)
  S(ba1(x)) -> ba0(S(x))
  plus(x,0()) -> x
  plus(0(),x) -> x
  plus(x,1()) -> S(x)
  plus(1(),x) -> S(x)
  plus(ba0(x),ba0(y)) -> plus(ba0(plus(x,y)),0())
  plus(ba0(x),ba1(y)) -> plus(ba0(plus(x,y)),1())
  plus(ba1(x),ba0(y)) -> plus(ba1(plus(x,y)),0())
  plus(ba1(x),ba1(y)) -> plus(ba1(plus(x,y)),1())
  mult(x,0()) -> 0()
  mult(x,1()) -> x
  mult(x,ba0(y)) -> plus(ba0(mult(x,y)),mult(x,0()))
  mult(x,ba1(y)) -> plus(ba0(mult(x,y)),mult(x,1()))
  critical peaks: joinable
   Reflection Processor:
    ba0(0()) -> 0()
    ba1(0()) -> 1()
    S(0()) -> 1()
    S(1()) -> ba0(1())
    S(ba0(x)) -> ba1(x)
    S(ba1(x)) -> ba0(S(x))
    plus(0(),x) -> x
    plus(x,0()) -> x
    plus(1(),x) -> S(x)
    plus(x,1()) -> S(x)
    plus(ba0(y),ba0(x)) -> plus(0(),ba0(plus(y,x)))
    plus(ba1(y),ba0(x)) -> plus(1(),ba0(plus(y,x)))
    plus(ba0(y),ba1(x)) -> plus(0(),ba1(plus(y,x)))
    plus(ba1(y),ba1(x)) -> plus(1(),ba1(plus(y,x)))
    mult(0(),x) -> 0()
    mult(1(),x) -> x
    mult(ba0(y),x) -> plus(mult(0(),x),ba0(mult(y,x)))
    mult(ba1(y),x) -> plus(mult(1(),x),ba0(mult(y,x)))
    LPO Processor:
     precedence:
      mult > plus > S > ba0 > ba1 > 1 ~ 0
     problem:
      
     Qed