YES

Problem 1: 

(VAR A B V1 V2 X Y)
(THEORY 
(AC mult plus union))
(RULES
0(z) -> z
U101(tt,X) -> X
U11(tt) -> z
U111(tt,A,B) -> plus(sum(A),sum(B))
U21(tt,X,Y) -> 0(mult(X,Y))
U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
U41(tt,X) -> X
U51(tt,X,Y) -> 0(plus(X,Y))
U61(tt,X,Y) -> 1(plus(X,Y))
U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
U81(tt,X) -> X
U91(tt,A,B) -> mult(prod(A),prod(B))
and(tt,X) -> X
isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
isBag(empty) -> tt
isBag(singl(V1)) -> isBin(V1)
isBin(0(V1)) -> isBin(V1)
isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
isBin(prod(V1)) -> isBag(V1)
isBin(sum(V1)) -> isBag(V1)
isBin(1(V1)) -> isBin(V1)
isBin(z) -> tt
mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
mult(z,X) -> U11(isBin(X))
plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
plus(z,X) -> U41(isBin(X),X)
prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
prod(empty) -> 1(z)
prod(singl(X)) -> U81(isBin(X),X)
sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
sum(empty) -> 0(z)
sum(singl(X)) -> U101(isBin(X),X)
union(empty,X) -> X
union(X,empty) -> X
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8))
 UNION(x6,x7) = UNION(x7,x6)
-> Pairs:
 U111#(tt,A,B) -> PLUS(sum(A),sum(B))
 U111#(tt,A,B) -> SUM(A)
 U111#(tt,A,B) -> SUM(B)
 U21#(tt,X,Y) -> 0#(mult(X,Y))
 U21#(tt,X,Y) -> MULT(X,Y)
 U31#(tt,X,Y) -> 0#(mult(X,Y))
 U31#(tt,X,Y) -> MULT(X,Y)
 U31#(tt,X,Y) -> PLUS(0(mult(X,Y)),Y)
 U51#(tt,X,Y) -> 0#(plus(X,Y))
 U51#(tt,X,Y) -> PLUS(X,Y)
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> 0#(plus(plus(X,Y),1(z)))
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 U91#(tt,A,B) -> MULT(prod(A),prod(B))
 U91#(tt,A,B) -> PROD(A)
 U91#(tt,A,B) -> PROD(B)
 ISBAG(union(V1,V2)) -> AND(isBag(V1),isBag(V2))
 ISBAG(union(V1,V2)) -> ISBAG(V1)
 ISBAG(union(V1,V2)) -> ISBAG(V2)
 ISBAG(singl(V1)) -> ISBIN(V1)
 ISBIN(0(V1)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> AND(isBin(V1),isBin(V2))
 ISBIN(mult(V1,V2)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> ISBIN(V2)
 ISBIN(plus(V1,V2)) -> AND(isBin(V1),isBin(V2))
 ISBIN(plus(V1,V2)) -> ISBIN(V1)
 ISBIN(plus(V1,V2)) -> ISBIN(V2)
 ISBIN(prod(V1)) -> ISBAG(V1)
 ISBIN(sum(V1)) -> ISBAG(V1)
 ISBIN(1(V1)) -> ISBIN(V1)
 MULT(0(X),Y) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(0(X),Y) -> AND(isBin(X),isBin(Y))
 MULT(0(X),Y) -> ISBIN(X)
 MULT(0(X),Y) -> ISBIN(Y)
 MULT(mult(0(X),Y),x6) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> AND(isBin(X),isBin(Y))
 MULT(mult(0(X),Y),x6) -> ISBIN(X)
 MULT(mult(0(X),Y),x6) -> ISBIN(Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> AND(isBin(X),isBin(Y))
 MULT(mult(1(X),Y),x6) -> ISBIN(X)
 MULT(mult(1(X),Y),x6) -> ISBIN(Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> U11#(isBin(X))
 MULT(mult(z,X),x6) -> ISBIN(X)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(1(X),Y) -> AND(isBin(X),isBin(Y))
 MULT(1(X),Y) -> ISBIN(X)
 MULT(1(X),Y) -> ISBIN(Y)
 MULT(z,X) -> U11#(isBin(X))
 MULT(z,X) -> ISBIN(X)
 PLUS(0(X),0(Y)) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),0(Y)) -> AND(isBin(X),isBin(Y))
 PLUS(0(X),0(Y)) -> ISBIN(X)
 PLUS(0(X),0(Y)) -> ISBIN(Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),1(Y)) -> AND(isBin(X),isBin(Y))
 PLUS(0(X),1(Y)) -> ISBIN(X)
 PLUS(0(X),1(Y)) -> ISBIN(Y)
 PLUS(plus(0(X),0(Y)),x6) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> AND(isBin(X),isBin(Y))
 PLUS(plus(0(X),0(Y)),x6) -> ISBIN(X)
 PLUS(plus(0(X),0(Y)),x6) -> ISBIN(Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> AND(isBin(X),isBin(Y))
 PLUS(plus(0(X),1(Y)),x6) -> ISBIN(X)
 PLUS(plus(0(X),1(Y)),x6) -> ISBIN(Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> AND(isBin(X),isBin(Y))
 PLUS(plus(1(X),1(Y)),x6) -> ISBIN(X)
 PLUS(plus(1(X),1(Y)),x6) -> ISBIN(Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> U41#(isBin(X),X)
 PLUS(plus(z,X),x6) -> ISBIN(X)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(1(X),1(Y)) -> AND(isBin(X),isBin(Y))
 PLUS(1(X),1(Y)) -> ISBIN(X)
 PLUS(1(X),1(Y)) -> ISBIN(Y)
 PLUS(z,X) -> U41#(isBin(X),X)
 PLUS(z,X) -> ISBIN(X)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
 PROD(union(A,B)) -> AND(isBag(A),isBag(B))
 PROD(union(A,B)) -> ISBAG(A)
 PROD(union(A,B)) -> ISBAG(B)
 PROD(singl(X)) -> U81#(isBin(X),X)
 PROD(singl(X)) -> ISBIN(X)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
 SUM(union(A,B)) -> AND(isBag(A),isBag(B))
 SUM(union(A,B)) -> ISBAG(A)
 SUM(union(A,B)) -> ISBAG(B)
 SUM(empty) -> 0#(z)
 SUM(singl(X)) -> U101#(isBin(X),X)
 SUM(singl(X)) -> ISBIN(X)
 UNION(union(empty,X),x6) -> UNION(X,x6)
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)

Problem 1: 

SCC Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8))
 UNION(x6,x7) = UNION(x7,x6)
-> Pairs:
 U111#(tt,A,B) -> PLUS(sum(A),sum(B))
 U111#(tt,A,B) -> SUM(A)
 U111#(tt,A,B) -> SUM(B)
 U21#(tt,X,Y) -> 0#(mult(X,Y))
 U21#(tt,X,Y) -> MULT(X,Y)
 U31#(tt,X,Y) -> 0#(mult(X,Y))
 U31#(tt,X,Y) -> MULT(X,Y)
 U31#(tt,X,Y) -> PLUS(0(mult(X,Y)),Y)
 U51#(tt,X,Y) -> 0#(plus(X,Y))
 U51#(tt,X,Y) -> PLUS(X,Y)
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> 0#(plus(plus(X,Y),1(z)))
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 U91#(tt,A,B) -> MULT(prod(A),prod(B))
 U91#(tt,A,B) -> PROD(A)
 U91#(tt,A,B) -> PROD(B)
 ISBAG(union(V1,V2)) -> AND(isBag(V1),isBag(V2))
 ISBAG(union(V1,V2)) -> ISBAG(V1)
 ISBAG(union(V1,V2)) -> ISBAG(V2)
 ISBAG(singl(V1)) -> ISBIN(V1)
 ISBIN(0(V1)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> AND(isBin(V1),isBin(V2))
 ISBIN(mult(V1,V2)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> ISBIN(V2)
 ISBIN(plus(V1,V2)) -> AND(isBin(V1),isBin(V2))
 ISBIN(plus(V1,V2)) -> ISBIN(V1)
 ISBIN(plus(V1,V2)) -> ISBIN(V2)
 ISBIN(prod(V1)) -> ISBAG(V1)
 ISBIN(sum(V1)) -> ISBAG(V1)
 ISBIN(1(V1)) -> ISBIN(V1)
 MULT(0(X),Y) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(0(X),Y) -> AND(isBin(X),isBin(Y))
 MULT(0(X),Y) -> ISBIN(X)
 MULT(0(X),Y) -> ISBIN(Y)
 MULT(mult(0(X),Y),x6) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> AND(isBin(X),isBin(Y))
 MULT(mult(0(X),Y),x6) -> ISBIN(X)
 MULT(mult(0(X),Y),x6) -> ISBIN(Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> AND(isBin(X),isBin(Y))
 MULT(mult(1(X),Y),x6) -> ISBIN(X)
 MULT(mult(1(X),Y),x6) -> ISBIN(Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> U11#(isBin(X))
 MULT(mult(z,X),x6) -> ISBIN(X)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(1(X),Y) -> AND(isBin(X),isBin(Y))
 MULT(1(X),Y) -> ISBIN(X)
 MULT(1(X),Y) -> ISBIN(Y)
 MULT(z,X) -> U11#(isBin(X))
 MULT(z,X) -> ISBIN(X)
 PLUS(0(X),0(Y)) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),0(Y)) -> AND(isBin(X),isBin(Y))
 PLUS(0(X),0(Y)) -> ISBIN(X)
 PLUS(0(X),0(Y)) -> ISBIN(Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),1(Y)) -> AND(isBin(X),isBin(Y))
 PLUS(0(X),1(Y)) -> ISBIN(X)
 PLUS(0(X),1(Y)) -> ISBIN(Y)
 PLUS(plus(0(X),0(Y)),x6) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> AND(isBin(X),isBin(Y))
 PLUS(plus(0(X),0(Y)),x6) -> ISBIN(X)
 PLUS(plus(0(X),0(Y)),x6) -> ISBIN(Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> AND(isBin(X),isBin(Y))
 PLUS(plus(0(X),1(Y)),x6) -> ISBIN(X)
 PLUS(plus(0(X),1(Y)),x6) -> ISBIN(Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> AND(isBin(X),isBin(Y))
 PLUS(plus(1(X),1(Y)),x6) -> ISBIN(X)
 PLUS(plus(1(X),1(Y)),x6) -> ISBIN(Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> U41#(isBin(X),X)
 PLUS(plus(z,X),x6) -> ISBIN(X)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(1(X),1(Y)) -> AND(isBin(X),isBin(Y))
 PLUS(1(X),1(Y)) -> ISBIN(X)
 PLUS(1(X),1(Y)) -> ISBIN(Y)
 PLUS(z,X) -> U41#(isBin(X),X)
 PLUS(z,X) -> ISBIN(X)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
 PROD(union(A,B)) -> AND(isBag(A),isBag(B))
 PROD(union(A,B)) -> ISBAG(A)
 PROD(union(A,B)) -> ISBAG(B)
 PROD(singl(X)) -> U81#(isBin(X),X)
 PROD(singl(X)) -> ISBIN(X)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
 SUM(union(A,B)) -> AND(isBag(A),isBag(B))
 SUM(union(A,B)) -> ISBAG(A)
 SUM(union(A,B)) -> ISBAG(B)
 SUM(empty) -> 0#(z)
 SUM(singl(X)) -> U101#(isBin(X),X)
 SUM(singl(X)) -> ISBIN(X)
 UNION(union(empty,X),x6) -> UNION(X,x6)
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 UNION(union(empty,X),x6) -> UNION(X,x6)
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 UNION(union(x6,x7),x8) -> UNION(x6,union(x7,x8))
 UNION(x6,x7) -> UNION(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)
->->Cycle:
->->-> Pairs:
 ISBAG(union(V1,V2)) -> ISBAG(V1)
 ISBAG(union(V1,V2)) -> ISBAG(V2)
 ISBAG(singl(V1)) -> ISBIN(V1)
 ISBIN(0(V1)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> ISBIN(V2)
 ISBIN(plus(V1,V2)) -> ISBIN(V1)
 ISBIN(plus(V1,V2)) -> ISBIN(V2)
 ISBIN(prod(V1)) -> ISBAG(V1)
 ISBIN(sum(V1)) -> ISBAG(V1)
 ISBIN(1(V1)) -> ISBIN(V1)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 U51#(tt,X,Y) -> PLUS(X,Y)
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(0(X),0(Y)) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->->Cycle:
->->-> Pairs:
 U21#(tt,X,Y) -> MULT(X,Y)
 U31#(tt,X,Y) -> MULT(X,Y)
 MULT(0(X),Y) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8))
 MULT(x6,x7) -> MULT(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->->Cycle:
->->-> Pairs:
 U91#(tt,A,B) -> PROD(A)
 U91#(tt,A,B) -> PROD(B)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 U111#(tt,A,B) -> SUM(A)
 U111#(tt,A,B) -> SUM(B)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty


The problem is decomposed in 6 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8))
 UNION(x6,x7) = UNION(x7,x6)
-> Pairs:
 UNION(union(empty,X),x6) -> UNION(X,x6)
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = 0
[U51](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U71](X1,X2,X3) = 0
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = 0
[isBag](X) = 0
[isBin](X) = 0
[mult](X1,X2) = 0
[plus](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[union](X1,X2) = X1 + X2
[1](X) = 0
[empty] = 2
[singl](X) = 0
[tt] = 0
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 2.X1 + 2.X2

Problem 1.1: 

SCC Processor:
-> FAxioms:
 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8))
 UNION(x6,x7) = UNION(x7,x6)
-> Pairs:
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 UNION(union(x6,x7),x8) -> UNION(x6,union(x7,x8))
 UNION(x6,x7) -> UNION(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8))
 UNION(x6,x7) = UNION(x7,x6)
-> Pairs:
 UNION(union(X,empty),x6) -> UNION(X,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = 0
[U51](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U71](X1,X2,X3) = 0
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = 0
[isBag](X) = 0
[isBin](X) = 0
[mult](X1,X2) = 0
[plus](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[union](X1,X2) = X1 + X2 + 2
[1](X) = 0
[empty] = 2
[singl](X) = 0
[tt] = 0
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 2.X1 + 2.X2

Problem 1.1: 

SCC Processor:
-> FAxioms:
 UNION(union(x6,x7),x8) = UNION(x6,union(x7,x8))
 UNION(x6,x7) = UNION(x7,x6)
-> Pairs:
 Empty
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 UNION(union(x6,x7),x8) -> UNION(x6,x7)
 UNION(x6,union(x7,x8)) -> UNION(x7,x8)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 ISBAG(union(V1,V2)) -> ISBAG(V1)
 ISBAG(union(V1,V2)) -> ISBAG(V2)
 ISBAG(singl(V1)) -> ISBIN(V1)
 ISBIN(0(V1)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> ISBIN(V1)
 ISBIN(mult(V1,V2)) -> ISBIN(V2)
 ISBIN(plus(V1,V2)) -> ISBIN(V1)
 ISBIN(plus(V1,V2)) -> ISBIN(V2)
 ISBIN(prod(V1)) -> ISBAG(V1)
 ISBIN(sum(V1)) -> ISBAG(V1)
 ISBIN(1(V1)) -> ISBIN(V1)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Projection:
 pi(ISBAG) = [1]
 pi(ISBIN) = [1]

Problem 1.2: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U51#(tt,X,Y) -> PLUS(X,Y)
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(0(X),0(Y)) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X2 + 1
[U51](X1,X2,X3) = X2 + X3 + 2
[U61](X1,X2,X3) = 2.X1 + X2 + X3
[U71](X1,X2,X3) = 2.X1 + X2 + X3 + 1
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = 2.X2
[plus](X1,X2) = X1 + X2 + 1
[prod](X) = 2.X + 2
[sum](X) = 2.X + 2
[union](X1,X2) = 2.X2 + 2
[1](X) = X + 2
[empty] = 1
[singl](X) = 2.X + 2
[tt] = 2
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U71#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(0(X),0(Y)) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> U51#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U61#(tt,X,Y) -> PLUS(X,Y)
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X2
[U51](X1,X2,X3) = X2 + X3 + 1
[U61](X1,X2,X3) = X2 + X3 + 1
[U71](X1,X2,X3) = X2 + X3 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2.X + 2
[isBin](X) = 2.X + 2
[mult](X1,X2) = 2.X1 + X2
[plus](X1,X2) = X1 + X2
[prod](X) = 2.X + 2
[sum](X) = 2.X + 2
[union](X1,X2) = 2.X1 + 2.X2 + 2
[1](X) = X + 1
[empty] = 2
[singl](X) = X + 2
[tt] = 2
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U71#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(0(X),1(Y)) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> U61#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U71#(tt,X,Y) -> PLUS(plus(X,Y),1(z))
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X2
[U51](X1,X2,X3) = X2 + X3
[U61](X1,X2,X3) = X1 + X2 + X3
[U71](X1,X2,X3) = X2 + X3 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = 2.X1 + 2.X2 + 1
[plus](X1,X2) = X1 + X2
[prod](X) = 2
[sum](X) = 1
[union](X1,X2) = 2.X1 + 2.X2 + 2
[1](X) = X + 2
[empty] = 2
[singl](X) = X + 1
[tt] = 2
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 U71#(tt,X,Y) -> PLUS(X,Y)
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X2 + 2
[U51](X1,X2,X3) = X2 + X3 + 2
[U61](X1,X2,X3) = X1 + X2 + X3 + 2
[U71](X1,X2,X3) = 2.X1 + X2 + X3 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = X1 + 2.X2 + 2
[plus](X1,X2) = X1 + X2 + 2
[prod](X) = 1
[sum](X) = X
[union](X1,X2) = 2
[1](X) = X + 2
[empty] = 2
[singl](X) = 0
[tt] = 2
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> U71#(and(isBin(X),isBin(Y)),X,Y)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
 PLUS(1(X),1(Y)) -> U71#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(0(X),0(Y)),x6) -> PLUS(U51(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X1 + X2 + 2
[U51](X1,X2,X3) = 2.X1 + 1
[U61](X1,X2,X3) = 2
[U71](X1,X2,X3) = 2.X1 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = 2.X1 + 2.X2 + 1
[plus](X1,X2) = X1 + X2 + 2
[prod](X) = 2.X
[sum](X) = 2.X + 2
[union](X1,X2) = 2.X1
[1](X) = 2
[empty] = 1
[singl](X) = 1
[tt] = 2
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(0(X),1(Y)),x6) -> PLUS(U61(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X2 + 2
[U51](X1,X2,X3) = 2
[U61](X1,X2,X3) = 2
[U71](X1,X2,X3) = 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = 2.X1 + 2.X2
[plus](X1,X2) = X1 + X2
[prod](X) = 2.X + 2
[sum](X) = 2.X + 2
[union](X1,X2) = 2.X1 + 2.X2
[1](X) = 2
[empty] = 0
[singl](X) = 2.X
[tt] = 2
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(1(X),1(Y)),x6) -> PLUS(U71(and(isBin(X),isBin(Y)),X,Y),x6)
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X2
[U51](X1,X2,X3) = X2 + X3
[U61](X1,X2,X3) = X2 + X3 + 2
[U71](X1,X2,X3) = X2 + X3 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2.X
[isBin](X) = 2.X + 2
[mult](X1,X2) = 2.X1 + 2.X2 + 2
[plus](X1,X2) = X1 + X2
[prod](X) = 2.X + 1
[sum](X) = X + 2
[union](X1,X2) = 2.X2 + 2
[1](X) = X + 2
[empty] = 2
[singl](X) = 2.X + 1
[tt] = 2
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 PLUS(plus(x6,x7),x8) -> PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) -> PLUS(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 PLUS(plus(z,X),x6) -> PLUS(U41(isBin(X),X),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = X1 + X2 + 1
[U51](X1,X2,X3) = 2
[U61](X1,X2,X3) = 2.X1 + 2
[U71](X1,X2,X3) = 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = X1 + 2.X2 + 1
[plus](X1,X2) = X1 + X2 + 2
[prod](X) = X + 2
[sum](X) = 2.X
[union](X1,X2) = 2
[1](X) = 2
[empty] = 0
[singl](X) = X + 1
[tt] = 0
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = X1 + X2
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x6,x7),x8) = PLUS(x6,plus(x7,x8))
 PLUS(x6,x7) = PLUS(x7,x6)
-> Pairs:
 Empty
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 PLUS(plus(x6,x7),x8) -> PLUS(x6,x7)
 PLUS(x6,plus(x7,x8)) -> PLUS(x7,x8)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 U21#(tt,X,Y) -> MULT(X,Y)
 U31#(tt,X,Y) -> MULT(X,Y)
 MULT(0(X),Y) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U11(tt) -> z
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X3 + 1
[U31](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X3 + 1
[U41](X1,X2) = X2
[U51](X1,X2,X3) = X1.X3 + X1 + X2 + 1
[U61](X1,X2,X3) = X1 + X2 + X3 + 1
[U71](X1,X2,X3) = X1.X2 + X1 + X3 + 1
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X1.X2
[isBag](X) = 1
[isBin](X) = 1
[mult](X1,X2) = X1.X2 + X1 + X2
[plus](X1,X2) = X1 + X2
[prod](X) = 1
[sum](X) = X.X + 1
[union](X1,X2) = X1 + X2 + 1
[1](X) = X + 1
[empty] = 1
[singl](X) = X.X
[tt] = 1
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = X1.X2.X3 + X1 + X2 + X3
[U31#](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = X1.X2 + X1 + X2
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 U31#(tt,X,Y) -> MULT(X,Y)
 MULT(0(X),Y) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> U21#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U31#(tt,X,Y) -> MULT(X,Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8))
 MULT(x6,x7) -> MULT(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 U31#(tt,X,Y) -> MULT(X,Y)
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U11(tt) -> z
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2
[U31](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3
[U41](X1,X2) = X1.X2
[U51](X1,X2,X3) = X1 + X2 + X3
[U61](X1,X2,X3) = X1.X3 + X1 + X2 + 1
[U71](X1,X2,X3) = X1.X3 + X1 + X2 + 1
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X1.X2
[isBag](X) = 1
[isBin](X) = 1
[mult](X1,X2) = X1.X2 + X1 + X2
[plus](X1,X2) = X1 + X2
[prod](X) = 1
[sum](X) = X.X + X + 1
[union](X1,X2) = X1.X2 + X1 + 1
[1](X) = X + 1
[empty] = 1
[singl](X) = X.X + X + 1
[tt] = 1
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 + 1
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = X1.X2 + X1 + X2 + 1
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> U31#(and(isBin(X),isBin(Y)),X,Y)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
 MULT(1(X),Y) -> U31#(and(isBin(X),isBin(Y)),X,Y)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8))
 MULT(x6,x7) -> MULT(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 MULT(mult(0(X),Y),x6) -> MULT(U21(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U11(tt) -> z
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[U101](X1,X2) = 0
[U11](X) = X + 1
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 2
[U31](X1,X2,X3) = X1 + X3 + 2
[U41](X1,X2) = 2.X1 + X2
[U51](X1,X2,X3) = 2.X1
[U61](X1,X2,X3) = 2.X1
[U71](X1,X2,X3) = 2.X1 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = X1 + X2 + 2
[plus](X1,X2) = X1 + X2 + 2
[prod](X) = 2
[sum](X) = X + 2
[union](X1,X2) = X1 + 2.X2 + 2
[1](X) = 2
[empty] = 1
[singl](X) = 2.X + 1
[tt] = 2
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 2.X1 + 2.X2
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8))
 MULT(x6,x7) -> MULT(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 MULT(mult(1(X),Y),x6) -> MULT(U31(and(isBin(X),isBin(Y)),X,Y),x6)
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U11(tt) -> z
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[U101](X1,X2) = 0
[U11](X) = X
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 1
[U31](X1,X2,X3) = X3 + 2
[U41](X1,X2) = X2 + 2
[U51](X1,X2,X3) = 0
[U61](X1,X2,X3) = X1 + 2
[U71](X1,X2,X3) = 2.X1 + 2
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = X1 + X2 + 2
[plus](X1,X2) = X1 + X2 + 2
[prod](X) = X + 2
[sum](X) = 2.X + 2
[union](X1,X2) = X1 + X2
[1](X) = 2
[empty] = 0
[singl](X) = 2.X + 2
[tt] = 2
[z] = 0
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = X1 + X2
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
 MULT(mult(x6,x7),x8) -> MULT(x6,mult(x7,x8))
 MULT(x6,x7) -> MULT(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 MULT(mult(z,X),x6) -> MULT(U11(isBin(X)),x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 0(z) -> z
 U11(tt) -> z
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[U101](X1,X2) = 0
[U11](X) = 2
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = X3 + 2
[U31](X1,X2,X3) = X1 + X3 + 2
[U41](X1,X2) = X2 + 2
[U51](X1,X2,X3) = 2.X1 + 1
[U61](X1,X2,X3) = 2.X1 + 2
[U71](X1,X2,X3) = 2.X1
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X2
[isBag](X) = 2
[isBin](X) = 2
[mult](X1,X2) = X1 + X2 + 2
[plus](X1,X2) = X1 + X2 + 2
[prod](X) = X + 1
[sum](X) = 2.X + 1
[union](X1,X2) = X1 + 2.X2 + 2
[1](X) = 2
[empty] = 1
[singl](X) = X + 2
[tt] = 2
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 2.X1 + 2.X2
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 MULT(mult(x6,x7),x8) = MULT(x6,mult(x7,x8))
 MULT(x6,x7) = MULT(x7,x6)
-> Pairs:
 Empty
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 MULT(mult(x6,x7),x8) -> MULT(x6,x7)
 MULT(x6,mult(x7,x8)) -> MULT(x7,x8)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Reduction Pairs Processor:
-> FAxioms:
 Empty
-> Pairs:
 U91#(tt,A,B) -> PROD(A)
 U91#(tt,A,B) -> PROD(B)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 Empty
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = 0
[U51](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U71](X1,X2,X3) = 0
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X1 + 2.X2 + 2
[isBag](X) = 2.X + 1
[isBin](X) = 2.X
[mult](X1,X2) = 2.X1 + 2.X2 + 1
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[prod](X) = 2.X + 2
[sum](X) = X + 2
[union](X1,X2) = 2.X1 + 2.X2 + 2
[1](X) = 2.X
[empty] = 2
[singl](X) = 2.X + 2
[tt] = 2
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 0
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 0
[PROD](X) = 2.X
[SUM](X) = 0
[UNION](X1,X2) = 0

Problem 1.5: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 U91#(tt,A,B) -> PROD(B)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U91#(tt,A,B) -> PROD(B)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty

Problem 1.5: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 U91#(tt,A,B) -> PROD(B)
 PROD(union(A,B)) -> U91#(and(isBag(A),isBag(B)),A,B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Projection:
 pi(U91#) = [3]
 pi(PROD) = [1]

Problem 1.5: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 U91#(tt,A,B) -> PROD(B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Reduction Pairs Processor:
-> FAxioms:
 Empty
-> Pairs:
 U111#(tt,A,B) -> SUM(A)
 U111#(tt,A,B) -> SUM(B)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Usable Equations:
 Empty
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> Usable Rules:
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X
[U101](X1,X2) = 0
[U11](X) = 0
[U111](X1,X2,X3) = 0
[U21](X1,X2,X3) = 0
[U31](X1,X2,X3) = 0
[U41](X1,X2) = 0
[U51](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U71](X1,X2,X3) = 0
[U81](X1,X2) = 0
[U91](X1,X2,X3) = 0
[and](X1,X2) = X1 + 2.X2 + 2
[isBag](X) = 2.X + 1
[isBin](X) = 2.X
[mult](X1,X2) = 2.X1 + 2.X2 + 1
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[prod](X) = 2.X + 2
[sum](X) = X + 2
[union](X1,X2) = 2.X1 + 2.X2 + 2
[1](X) = 2.X
[empty] = 2
[singl](X) = 2.X + 2
[tt] = 2
[z] = 2
[0#](X) = 0
[U101#](X1,X2) = 0
[U11#](X) = 0
[U111#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U21#](X1,X2,X3) = 0
[U31#](X1,X2,X3) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U71#](X1,X2,X3) = 0
[U81#](X1,X2) = 0
[U91#](X1,X2,X3) = 0
[AND](X1,X2) = 0
[ISBAG](X) = 0
[ISBIN](X) = 0
[MULT](X1,X2) = 0
[PLUS](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 2.X
[UNION](X1,X2) = 0

Problem 1.6: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 U111#(tt,A,B) -> SUM(B)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U111#(tt,A,B) -> SUM(B)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
-> FAxioms:
 mult(mult(x6,x7),x8) -> mult(x6,mult(x7,x8))
 mult(x6,x7) -> mult(x7,x6)
 plus(plus(x6,x7),x8) -> plus(x6,plus(x7,x8))
 plus(x6,x7) -> plus(x7,x6)
 union(union(x6,x7),x8) -> union(x6,union(x7,x8))
 union(x6,x7) -> union(x7,x6)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
->->-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty

Problem 1.6: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 U111#(tt,A,B) -> SUM(B)
 SUM(union(A,B)) -> U111#(and(isBag(A),isBag(B)),A,B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Projection:
 pi(U111#) = [3]
 pi(SUM) = [1]

Problem 1.6: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 U111#(tt,A,B) -> SUM(B)
-> EAxioms:
 mult(mult(x6,x7),x8) = mult(x6,mult(x7,x8))
 mult(x6,x7) = mult(x7,x6)
 plus(plus(x6,x7),x8) = plus(x6,plus(x7,x8))
 plus(x6,x7) = plus(x7,x6)
 union(union(x6,x7),x8) = union(x6,union(x7,x8))
 union(x6,x7) = union(x7,x6)
-> Rules:
 0(z) -> z
 U101(tt,X) -> X
 U11(tt) -> z
 U111(tt,A,B) -> plus(sum(A),sum(B))
 U21(tt,X,Y) -> 0(mult(X,Y))
 U31(tt,X,Y) -> plus(0(mult(X,Y)),Y)
 U41(tt,X) -> X
 U51(tt,X,Y) -> 0(plus(X,Y))
 U61(tt,X,Y) -> 1(plus(X,Y))
 U71(tt,X,Y) -> 0(plus(plus(X,Y),1(z)))
 U81(tt,X) -> X
 U91(tt,A,B) -> mult(prod(A),prod(B))
 and(tt,X) -> X
 isBag(union(V1,V2)) -> and(isBag(V1),isBag(V2))
 isBag(empty) -> tt
 isBag(singl(V1)) -> isBin(V1)
 isBin(0(V1)) -> isBin(V1)
 isBin(mult(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(plus(V1,V2)) -> and(isBin(V1),isBin(V2))
 isBin(prod(V1)) -> isBag(V1)
 isBin(sum(V1)) -> isBag(V1)
 isBin(1(V1)) -> isBin(V1)
 isBin(z) -> tt
 mult(0(X),Y) -> U21(and(isBin(X),isBin(Y)),X,Y)
 mult(1(X),Y) -> U31(and(isBin(X),isBin(Y)),X,Y)
 mult(z,X) -> U11(isBin(X))
 plus(0(X),0(Y)) -> U51(and(isBin(X),isBin(Y)),X,Y)
 plus(0(X),1(Y)) -> U61(and(isBin(X),isBin(Y)),X,Y)
 plus(1(X),1(Y)) -> U71(and(isBin(X),isBin(Y)),X,Y)
 plus(z,X) -> U41(isBin(X),X)
 prod(union(A,B)) -> U91(and(isBag(A),isBag(B)),A,B)
 prod(empty) -> 1(z)
 prod(singl(X)) -> U81(isBin(X),X)
 sum(union(A,B)) -> U111(and(isBag(A),isBag(B)),A,B)
 sum(empty) -> 0(z)
 sum(singl(X)) -> U101(isBin(X),X)
 union(empty,X) -> X
 union(X,empty) -> X
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.