(COMMENT 

  Claude Marché

  Sum and product of a bag of integers

  Binary notation

)

(VAR b  y  x)
(THEORY (AC U * +))

(RULES

0(#) -> # 
+(#,x) -> x 
+(0(x),0(y)) -> 0(+(x,y)) 
+(0(x),1(y)) -> 1(+(x,y)) 
+(1(x),1(y)) -> 0(+(1(#),+(x,y))) 
*(#,x) -> # 
*(0(x),y) -> 0(*(x,y)) 
*(1(x),y) -> +(0(*(x,y)),y) 
U(empty,b) -> b 
sum(empty) -> 0(#) 
sum(singl(x)) -> x 
sum(U(x,y)) -> +(sum(x),sum(y)) 
prod(empty) -> 1(#) 
prod(singl(x)) -> x 
prod(U(x,y)) -> *(prod(x),prod(y)) 

)