2.59/3.15	YES
2.59/3.15	
2.59/3.15	Problem 1: 
2.59/3.15	
2.59/3.15	(VAR A B X Y)
2.59/3.15	(THEORY 
2.59/3.15	(AC mult plus union))
2.59/3.15	(RULES
2.59/3.15	0(z) -> z
2.59/3.15	and(tt,X) -> X
2.59/3.15	mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.15	mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.15	mult(z,X) -> z
2.59/3.15	plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.15	plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.15	plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.15	plus(z,X) -> X
2.59/3.15	prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.15	prod(empty) -> 1(z)
2.59/3.15	prod(singl(X)) -> X
2.59/3.15	sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.15	sum(empty) -> 0(z)
2.59/3.15	sum(singl(X)) -> X
2.59/3.15	union(empty,X) -> X
2.59/3.15	union(X,empty) -> X
2.59/3.15	)
2.59/3.15	
2.59/3.15	Problem 1: 
2.59/3.15	
2.59/3.15	Dependency Pairs Processor:
2.59/3.15	-> FAxioms:
2.59/3.15	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.15	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.15	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.15	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.15	 UNION(union(x4,x5),x6) = UNION(x4,union(x5,x6))
2.59/3.15	 UNION(x4,x5) = UNION(x5,x4)
2.59/3.15	-> Pairs:
2.59/3.15	 MULT(0(X),Y) -> 0#(mult(X,Y))
2.59/3.15	 MULT(0(X),Y) -> MULT(X,Y)
2.59/3.15	 MULT(mult(0(X),Y),x4) -> 0#(mult(X,Y))
2.59/3.15	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.15	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.15	 MULT(mult(1(X),Y),x4) -> 0#(mult(X,Y))
2.59/3.15	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.15	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.15	 MULT(mult(1(X),Y),x4) -> PLUS(0(mult(X,Y)),Y)
2.59/3.15	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.15	 MULT(1(X),Y) -> 0#(mult(X,Y))
2.59/3.15	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.15	 MULT(1(X),Y) -> PLUS(0(mult(X,Y)),Y)
2.59/3.15	 PLUS(0(X),0(Y)) -> 0#(plus(X,Y))
2.59/3.15	 PLUS(0(X),0(Y)) -> PLUS(X,Y)
2.59/3.15	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> 0#(plus(X,Y))
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.15	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> 0#(plus(plus(X,Y),1(z)))
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.15	 PLUS(1(X),1(Y)) -> 0#(plus(plus(X,Y),1(z)))
2.59/3.15	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.15	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.15	 PROD(union(A,B)) -> MULT(prod(A),prod(B))
2.59/3.15	 PROD(union(A,B)) -> PROD(A)
2.59/3.15	 PROD(union(A,B)) -> PROD(B)
2.59/3.15	 SUM(union(A,B)) -> PLUS(sum(A),sum(B))
2.59/3.15	 SUM(union(A,B)) -> SUM(A)
2.59/3.15	 SUM(union(A,B)) -> SUM(B)
2.59/3.15	 SUM(empty) -> 0#(z)
2.59/3.15	 UNION(union(empty,X),x4) -> UNION(X,x4)
2.59/3.15	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.15	-> EAxioms:
2.59/3.15	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.15	 mult(x4,x5) = mult(x5,x4)
2.59/3.15	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.15	 plus(x4,x5) = plus(x5,x4)
2.59/3.15	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.15	 union(x4,x5) = union(x5,x4)
2.59/3.15	-> Rules:
2.59/3.15	 0(z) -> z
2.59/3.15	 and(tt,X) -> X
2.59/3.15	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.15	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.15	 mult(z,X) -> z
2.59/3.15	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.15	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.15	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.15	 plus(z,X) -> X
2.59/3.15	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.15	 prod(empty) -> 1(z)
2.59/3.15	 prod(singl(X)) -> X
2.59/3.15	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.15	 sum(empty) -> 0(z)
2.59/3.15	 sum(singl(X)) -> X
2.59/3.15	 union(empty,X) -> X
2.59/3.15	 union(X,empty) -> X
2.59/3.15	-> SRules:
2.59/3.15	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.15	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.15	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.15	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.15	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.15	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.15	
2.59/3.15	Problem 1: 
2.59/3.15	
2.59/3.15	SCC Processor:
2.59/3.15	-> FAxioms:
2.59/3.15	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.15	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.15	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.15	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.15	 UNION(union(x4,x5),x6) = UNION(x4,union(x5,x6))
2.59/3.15	 UNION(x4,x5) = UNION(x5,x4)
2.59/3.15	-> Pairs:
2.59/3.15	 MULT(0(X),Y) -> 0#(mult(X,Y))
2.59/3.15	 MULT(0(X),Y) -> MULT(X,Y)
2.59/3.15	 MULT(mult(0(X),Y),x4) -> 0#(mult(X,Y))
2.59/3.15	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.15	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.15	 MULT(mult(1(X),Y),x4) -> 0#(mult(X,Y))
2.59/3.15	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.15	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.15	 MULT(mult(1(X),Y),x4) -> PLUS(0(mult(X,Y)),Y)
2.59/3.15	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.15	 MULT(1(X),Y) -> 0#(mult(X,Y))
2.59/3.15	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.15	 MULT(1(X),Y) -> PLUS(0(mult(X,Y)),Y)
2.59/3.15	 PLUS(0(X),0(Y)) -> 0#(plus(X,Y))
2.59/3.15	 PLUS(0(X),0(Y)) -> PLUS(X,Y)
2.59/3.15	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> 0#(plus(X,Y))
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.15	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> 0#(plus(plus(X,Y),1(z)))
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.15	 PLUS(1(X),1(Y)) -> 0#(plus(plus(X,Y),1(z)))
2.59/3.15	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.15	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.15	 PROD(union(A,B)) -> MULT(prod(A),prod(B))
2.59/3.15	 PROD(union(A,B)) -> PROD(A)
2.59/3.15	 PROD(union(A,B)) -> PROD(B)
2.59/3.15	 SUM(union(A,B)) -> PLUS(sum(A),sum(B))
2.59/3.15	 SUM(union(A,B)) -> SUM(A)
2.59/3.15	 SUM(union(A,B)) -> SUM(B)
2.59/3.15	 SUM(empty) -> 0#(z)
2.59/3.15	 UNION(union(empty,X),x4) -> UNION(X,x4)
2.59/3.15	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.15	-> EAxioms:
2.59/3.15	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.15	 mult(x4,x5) = mult(x5,x4)
2.59/3.15	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.15	 plus(x4,x5) = plus(x5,x4)
2.59/3.15	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.15	 union(x4,x5) = union(x5,x4)
2.59/3.15	-> Rules:
2.59/3.15	 0(z) -> z
2.59/3.15	 and(tt,X) -> X
2.59/3.15	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.15	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.15	 mult(z,X) -> z
2.59/3.15	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.15	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.15	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.15	 plus(z,X) -> X
2.59/3.15	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.15	 prod(empty) -> 1(z)
2.59/3.15	 prod(singl(X)) -> X
2.59/3.15	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.15	 sum(empty) -> 0(z)
2.59/3.15	 sum(singl(X)) -> X
2.59/3.15	 union(empty,X) -> X
2.59/3.15	 union(X,empty) -> X
2.59/3.15	-> SRules:
2.59/3.15	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.15	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.15	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.15	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.15	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.15	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.15	->Strongly Connected Components:
2.59/3.15	->->Cycle:
2.59/3.15	->->-> Pairs:
2.59/3.15	 UNION(union(empty,X),x4) -> UNION(X,x4)
2.59/3.15	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.15	-> FAxioms:
2.59/3.15	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.15	 mult(x4,x5) -> mult(x5,x4)
2.59/3.15	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.15	 plus(x4,x5) -> plus(x5,x4)
2.59/3.15	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.15	 union(x4,x5) -> union(x5,x4)
2.59/3.15	 UNION(union(x4,x5),x6) -> UNION(x4,union(x5,x6))
2.59/3.15	 UNION(x4,x5) -> UNION(x5,x4)
2.59/3.15	-> EAxioms:
2.59/3.15	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.15	 mult(x4,x5) = mult(x5,x4)
2.59/3.15	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.15	 plus(x4,x5) = plus(x5,x4)
2.59/3.15	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.15	 union(x4,x5) = union(x5,x4)
2.59/3.15	->->-> Rules:
2.59/3.15	 0(z) -> z
2.59/3.15	 and(tt,X) -> X
2.59/3.15	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.15	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.15	 mult(z,X) -> z
2.59/3.15	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.15	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.15	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.15	 plus(z,X) -> X
2.59/3.15	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.15	 prod(empty) -> 1(z)
2.59/3.15	 prod(singl(X)) -> X
2.59/3.15	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.15	 sum(empty) -> 0(z)
2.59/3.15	 sum(singl(X)) -> X
2.59/3.15	 union(empty,X) -> X
2.59/3.15	 union(X,empty) -> X
2.59/3.15	-> SRules:
2.59/3.15	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.15	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.15	->->Cycle:
2.59/3.15	->->-> Pairs:
2.59/3.15	 PLUS(0(X),0(Y)) -> PLUS(X,Y)
2.59/3.15	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.15	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.15	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.15	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.15	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.15	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.15	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.15	-> FAxioms:
2.59/3.15	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.15	 mult(x4,x5) -> mult(x5,x4)
2.59/3.15	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.15	 plus(x4,x5) -> plus(x5,x4)
2.59/3.15	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.15	 union(x4,x5) -> union(x5,x4)
2.59/3.15	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.15	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.15	-> EAxioms:
2.59/3.15	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.15	 mult(x4,x5) = mult(x5,x4)
2.59/3.15	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.15	 plus(x4,x5) = plus(x5,x4)
2.59/3.15	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.15	 union(x4,x5) = union(x5,x4)
2.59/3.15	->->-> Rules:
2.59/3.15	 0(z) -> z
2.59/3.15	 and(tt,X) -> X
2.59/3.15	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 SUM(union(A,B)) -> SUM(A)
2.59/3.16	 SUM(union(A,B)) -> SUM(B)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 Empty
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 MULT(0(X),Y) -> MULT(X,Y)
2.59/3.16	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.16	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.16	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.16	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.16	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.16	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.16	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.16	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PROD(union(A,B)) -> PROD(A)
2.59/3.16	 PROD(union(A,B)) -> PROD(B)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 Empty
2.59/3.16	
2.59/3.16	
2.59/3.16	The problem is decomposed in 5 subproblems.
2.59/3.16	
2.59/3.16	Problem 1.1: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 UNION(union(x4,x5),x6) = UNION(x4,union(x5,x6))
2.59/3.16	 UNION(x4,x5) = UNION(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 UNION(union(empty,X),x4) -> UNION(X,x4)
2.59/3.16	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.16	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = 0
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = 0
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = X1 + X2
2.59/3.16	[1](X) = 0
2.59/3.16	[empty] = 2
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 0
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 2.X1 + 2.X2
2.59/3.16	
2.59/3.16	Problem 1.1: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 UNION(union(x4,x5),x6) = UNION(x4,union(x5,x6))
2.59/3.16	 UNION(x4,x5) = UNION(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.16	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 UNION(union(x4,x5),x6) -> UNION(x4,union(x5,x6))
2.59/3.16	 UNION(x4,x5) -> UNION(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.16	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.1: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 UNION(union(x4,x5),x6) = UNION(x4,union(x5,x6))
2.59/3.16	 UNION(x4,x5) = UNION(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 UNION(union(X,empty),x4) -> UNION(X,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.16	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = 0
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = 0
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = X1 + X2 + 2
2.59/3.16	[1](X) = 0
2.59/3.16	[empty] = 2
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 0
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 2.X1 + 2.X2
2.59/3.16	
2.59/3.16	Problem 1.1: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 UNION(union(x4,x5),x6) = UNION(x4,union(x5,x6))
2.59/3.16	 UNION(x4,x5) = UNION(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 Empty
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 UNION(union(x4,x5),x6) -> UNION(x4,x5)
2.59/3.16	 UNION(x4,union(x5,x6)) -> UNION(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	 There is no strongly connected component
2.59/3.16	
2.59/3.16	The problem is finite.
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(0(X),0(Y)) -> PLUS(X,Y)
2.59/3.16	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X + 1
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 1
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(0(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X + 1
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 1
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(0(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X + 2
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = X1 + X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),0(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X + 1
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 1
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(1(plus(X,Y)),x4)
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X + 1
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 1
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(0(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = 2.X
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = 2.X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(0(plus(plus(X,Y),1(z))),x4)
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 1
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 2
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(1(X),1(Y)),x4) -> PLUS(X,Y)
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 2
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 0
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(plus(z,X),x4) -> PLUS(X,x4)
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> Usable Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Interpretation type:
2.59/3.16	 Linear
2.59/3.16	->Coefficients:
2.59/3.16	 Natural Numbers
2.59/3.16	->Dimension:
2.59/3.16	 1
2.59/3.16	->Bound:
2.59/3.16	 2
2.59/3.16	->Interpretation:
2.59/3.16	 
2.59/3.16	[0](X) = X
2.59/3.16	[and](X1,X2) = 0
2.59/3.16	[mult](X1,X2) = 0
2.59/3.16	[plus](X1,X2) = X1 + X2 + 1
2.59/3.16	[prod](X) = 0
2.59/3.16	[sum](X) = 0
2.59/3.16	[union](X1,X2) = 0
2.59/3.16	[1](X) = X + 2
2.59/3.16	[empty] = 0
2.59/3.16	[singl](X) = 0
2.59/3.16	[tt] = 0
2.59/3.16	[z] = 1
2.59/3.16	[0#](X) = 0
2.59/3.16	[MULT](X1,X2) = 0
2.59/3.16	[PLUS](X1,X2) = X1 + X2
2.59/3.16	[PROD](X) = 0
2.59/3.16	[SUM](X) = 0
2.59/3.16	[UNION](X1,X2) = 0
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	SCC Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	->Strongly Connected Components:
2.59/3.16	->->Cycle:
2.59/3.16	->->-> Pairs:
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> FAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) -> mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) -> plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) -> union(x5,x4)
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	->->-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.16	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.16	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.16	 mult(z,X) -> z
2.59/3.16	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.16	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.16	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.16	 plus(z,X) -> X
2.59/3.16	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.16	 prod(empty) -> 1(z)
2.59/3.16	 prod(singl(X)) -> X
2.59/3.16	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.16	 sum(empty) -> 0(z)
2.59/3.16	 sum(singl(X)) -> X
2.59/3.16	 union(empty,X) -> X
2.59/3.16	 union(X,empty) -> X
2.59/3.16	-> SRules:
2.59/3.16	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.16	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.16	
2.59/3.16	Problem 1.2: 
2.59/3.16	
2.59/3.16	Reduction Pairs Processor:
2.59/3.16	-> FAxioms:
2.59/3.16	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.16	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.16	-> Pairs:
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.16	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.16	-> EAxioms:
2.59/3.16	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.16	 mult(x4,x5) = mult(x5,x4)
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.16	 union(x4,x5) = union(x5,x4)
2.59/3.16	-> Usable Equations:
2.59/3.16	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.16	 plus(x4,x5) = plus(x5,x4)
2.59/3.16	-> Rules:
2.59/3.16	 0(z) -> z
2.59/3.16	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Linear
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 2
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = 0
2.59/3.17	[plus](X1,X2) = X1 + X2 + 2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 2
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 0
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = 0
2.59/3.17	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.2: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.2: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(plus(X,Y),1(z))
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Linear
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 2
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = 0
2.59/3.17	[plus](X1,X2) = X1 + X2 + 1
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 2
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 0
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = 0
2.59/3.17	[PLUS](X1,X2) = X1 + X2
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.2: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) -> PLUS(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.2: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 PLUS(1(X),1(Y)) -> PLUS(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Linear
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 2
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = 0
2.59/3.17	[plus](X1,X2) = X1 + X2 + 1
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 2
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 1
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = 0
2.59/3.17	[PLUS](X1,X2) = 2.X1 + 2.X2
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.2: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
2.59/3.17	 PLUS(x4,x5) = PLUS(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 Empty
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	 There is no strongly connected component
2.59/3.17	
2.59/3.17	The problem is finite.
2.59/3.17	
2.59/3.17	Problem 1.3: 
2.59/3.17	
2.59/3.17	Subterm Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 Empty
2.59/3.17	-> Pairs:
2.59/3.17	 SUM(union(A,B)) -> SUM(A)
2.59/3.17	 SUM(union(A,B)) -> SUM(B)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 Empty
2.59/3.17	->Projection:
2.59/3.17	 pi(SUM) = [1]
2.59/3.17	
2.59/3.17	Problem 1.3: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 Empty
2.59/3.17	-> Pairs:
2.59/3.17	 Empty
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 Empty
2.59/3.17	->Strongly Connected Components:
2.59/3.17	 There is no strongly connected component
2.59/3.17	
2.59/3.17	The problem is finite.
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(0(X),Y) -> MULT(X,Y)
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Simple mixed
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 1
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X + 1
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[plus](X1,X2) = X1 + X2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 1
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 0
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Simple mixed
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 1
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X + 1
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[plus](X1,X2) = X1 + X2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 1
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 0
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(plus(0(mult(X,Y)),Y),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Simple mixed
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 1
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[plus](X1,X2) = X1 + X2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 1
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 1
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(1(X),Y),x4) -> MULT(X,Y)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Simple mixed
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 1
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[plus](X1,X2) = X1 + X2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 1
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 1
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	 MULT(1(X),Y) -> MULT(X,Y)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Simple mixed
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 1
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = X
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[plus](X1,X2) = X1 + X2 + 1
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = X + 1
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 0
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = X1.X2 + X1 + X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(0(X),Y),x4) -> MULT(0(mult(X,Y)),x4)
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Linear
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 2
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = 2
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1 + X2 + 2
2.59/3.17	[plus](X1,X2) = X1 + X2 + 2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = 2
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 0
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = 2.X1 + 2.X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	->->Cycle:
2.59/3.17	->->-> Pairs:
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	-> FAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) -> mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) -> mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) -> plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) -> union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) -> union(x5,x4)
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) -> MULT(x5,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	->->-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	Reduction Pairs Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 MULT(mult(z,X),x4) -> MULT(z,x4)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Usable Equations:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> Usable Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Interpretation type:
2.59/3.17	 Linear
2.59/3.17	->Coefficients:
2.59/3.17	 Natural Numbers
2.59/3.17	->Dimension:
2.59/3.17	 1
2.59/3.17	->Bound:
2.59/3.17	 2
2.59/3.17	->Interpretation:
2.59/3.17	 
2.59/3.17	[0](X) = 2
2.59/3.17	[and](X1,X2) = 0
2.59/3.17	[mult](X1,X2) = X1 + X2 + 2
2.59/3.17	[plus](X1,X2) = X1 + X2 + 2
2.59/3.17	[prod](X) = 0
2.59/3.17	[sum](X) = 0
2.59/3.17	[union](X1,X2) = 0
2.59/3.17	[1](X) = 2
2.59/3.17	[empty] = 0
2.59/3.17	[singl](X) = 0
2.59/3.17	[tt] = 0
2.59/3.17	[z] = 2
2.59/3.17	[0#](X) = 0
2.59/3.17	[MULT](X1,X2) = 2.X1 + 2.X2
2.59/3.17	[PLUS](X1,X2) = 0
2.59/3.17	[PROD](X) = 0
2.59/3.17	[SUM](X) = 0
2.59/3.17	[UNION](X1,X2) = 0
2.59/3.17	
2.59/3.17	Problem 1.4: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 MULT(mult(x4,x5),x6) = MULT(x4,mult(x5,x6))
2.59/3.17	 MULT(x4,x5) = MULT(x5,x4)
2.59/3.17	-> Pairs:
2.59/3.17	 Empty
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 MULT(mult(x4,x5),x6) -> MULT(x4,x5)
2.59/3.17	 MULT(x4,mult(x5,x6)) -> MULT(x5,x6)
2.59/3.17	->Strongly Connected Components:
2.59/3.17	 There is no strongly connected component
2.59/3.17	
2.59/3.17	The problem is finite.
2.59/3.17	
2.59/3.17	Problem 1.5: 
2.59/3.17	
2.59/3.17	Subterm Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 Empty
2.59/3.17	-> Pairs:
2.59/3.17	 PROD(union(A,B)) -> PROD(A)
2.59/3.17	 PROD(union(A,B)) -> PROD(B)
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 Empty
2.59/3.17	->Projection:
2.59/3.17	 pi(PROD) = [1]
2.59/3.17	
2.59/3.17	Problem 1.5: 
2.59/3.17	
2.59/3.17	SCC Processor:
2.59/3.17	-> FAxioms:
2.59/3.17	 Empty
2.59/3.17	-> Pairs:
2.59/3.17	 Empty
2.59/3.17	-> EAxioms:
2.59/3.17	 mult(mult(x4,x5),x6) = mult(x4,mult(x5,x6))
2.59/3.17	 mult(x4,x5) = mult(x5,x4)
2.59/3.17	 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
2.59/3.17	 plus(x4,x5) = plus(x5,x4)
2.59/3.17	 union(union(x4,x5),x6) = union(x4,union(x5,x6))
2.59/3.17	 union(x4,x5) = union(x5,x4)
2.59/3.17	-> Rules:
2.59/3.17	 0(z) -> z
2.59/3.17	 and(tt,X) -> X
2.59/3.17	 mult(0(X),Y) -> 0(mult(X,Y))
2.59/3.17	 mult(1(X),Y) -> plus(0(mult(X,Y)),Y)
2.59/3.17	 mult(z,X) -> z
2.59/3.17	 plus(0(X),0(Y)) -> 0(plus(X,Y))
2.59/3.17	 plus(0(X),1(Y)) -> 1(plus(X,Y))
2.59/3.17	 plus(1(X),1(Y)) -> 0(plus(plus(X,Y),1(z)))
2.59/3.17	 plus(z,X) -> X
2.59/3.17	 prod(union(A,B)) -> mult(prod(A),prod(B))
2.59/3.17	 prod(empty) -> 1(z)
2.59/3.17	 prod(singl(X)) -> X
2.59/3.17	 sum(union(A,B)) -> plus(sum(A),sum(B))
2.59/3.17	 sum(empty) -> 0(z)
2.59/3.17	 sum(singl(X)) -> X
2.59/3.17	 union(empty,X) -> X
2.59/3.17	 union(X,empty) -> X
2.59/3.17	-> SRules:
2.59/3.17	 Empty
2.59/3.17	->Strongly Connected Components:
2.59/3.17	 There is no strongly connected component
2.59/3.17	
2.59/3.17	The problem is finite.
2.59/3.17	EOF