16.60/17.56	YES
16.60/17.56	
16.60/17.56	Problem 1: 
16.60/17.56	
16.60/17.56	(VAR b x y z)
16.60/17.56	(THEORY 
16.60/17.56	(AC * + U))
16.60/17.56	(RULES
16.60/17.56	*(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	*(0(x),y) -> 0(*(x,y))
16.60/17.56	*(#,x) -> #
16.60/17.56	*(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	+(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	+(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	+(#,x) -> x
16.60/17.56	+(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	0(#) -> #
16.60/17.56	U(empty,b) -> b
16.60/17.56	prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	prod(empty) -> 1(#)
16.60/17.56	prod(singl(x)) -> x
16.60/17.56	sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	sum(empty) -> 0(#)
16.60/17.56	sum(singl(x)) -> x
16.60/17.56	)
16.60/17.56	
16.60/17.56	Problem 1: 
16.60/17.56	
16.60/17.56	Dependency Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.56	 *#(x4,x5) = *#(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	 U#(U(x4,x5),x6) = U#(x4,U(x5,x6))
16.60/17.56	 U#(x4,x5) = U#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(x,y)
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.56	 *#(*(+(y,z),x),x4) -> +#(*(x,y),*(x,z))
16.60/17.56	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.56	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.56	 *#(*(0(x),y),x4) -> 0#(*(x,y))
16.60/17.56	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.56	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.56	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.56	 *#(*(1(x),y),x4) -> +#(0(*(x,y)),y)
16.60/17.56	 *#(*(1(x),y),x4) -> 0#(*(x,y))
16.60/17.56	 *#(+(y,z),x) -> *#(x,y)
16.60/17.56	 *#(+(y,z),x) -> *#(x,z)
16.60/17.56	 *#(+(y,z),x) -> +#(*(x,y),*(x,z))
16.60/17.56	 *#(0(x),y) -> *#(x,y)
16.60/17.56	 *#(0(x),y) -> 0#(*(x,y))
16.60/17.56	 *#(1(x),y) -> *#(x,y)
16.60/17.56	 *#(1(x),y) -> +#(0(*(x,y)),y)
16.60/17.56	 *#(1(x),y) -> 0#(*(x,y))
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(0(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),0(y)),x4) -> 0#(+(x,y))
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> 0#(+(1(#),+(x,y)))
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> 0#(+(x,y))
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> 0#(+(1(#),+(x,y)))
16.60/17.56	 U#(U(empty,b),x4) -> U#(b,x4)
16.60/17.56	 PROD(U(x,y)) -> *#(prod(x),prod(y))
16.60/17.56	 PROD(U(x,y)) -> PROD(x)
16.60/17.56	 PROD(U(x,y)) -> PROD(y)
16.60/17.56	 SUM(U(x,y)) -> +#(sum(x),sum(y))
16.60/17.56	 SUM(U(x,y)) -> SUM(x)
16.60/17.56	 SUM(U(x,y)) -> SUM(y)
16.60/17.56	 SUM(empty) -> 0#(#)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.56	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	 U#(U(x4,x5),x6) -> U#(x4,x5)
16.60/17.56	 U#(x4,U(x5,x6)) -> U#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.56	 *#(x4,x5) = *#(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	 U#(U(x4,x5),x6) = U#(x4,U(x5,x6))
16.60/17.56	 U#(x4,x5) = U#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(x,y)
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.56	 *#(*(+(y,z),x),x4) -> +#(*(x,y),*(x,z))
16.60/17.56	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.56	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.56	 *#(*(0(x),y),x4) -> 0#(*(x,y))
16.60/17.56	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.56	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.56	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.56	 *#(*(1(x),y),x4) -> +#(0(*(x,y)),y)
16.60/17.56	 *#(*(1(x),y),x4) -> 0#(*(x,y))
16.60/17.56	 *#(+(y,z),x) -> *#(x,y)
16.60/17.56	 *#(+(y,z),x) -> *#(x,z)
16.60/17.56	 *#(+(y,z),x) -> +#(*(x,y),*(x,z))
16.60/17.56	 *#(0(x),y) -> *#(x,y)
16.60/17.56	 *#(0(x),y) -> 0#(*(x,y))
16.60/17.56	 *#(1(x),y) -> *#(x,y)
16.60/17.56	 *#(1(x),y) -> +#(0(*(x,y)),y)
16.60/17.56	 *#(1(x),y) -> 0#(*(x,y))
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(0(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),0(y)),x4) -> 0#(+(x,y))
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> 0#(+(1(#),+(x,y)))
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> 0#(+(x,y))
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> 0#(+(1(#),+(x,y)))
16.60/17.56	 U#(U(empty,b),x4) -> U#(b,x4)
16.60/17.56	 PROD(U(x,y)) -> *#(prod(x),prod(y))
16.60/17.56	 PROD(U(x,y)) -> PROD(x)
16.60/17.56	 PROD(U(x,y)) -> PROD(y)
16.60/17.56	 SUM(U(x,y)) -> +#(sum(x),sum(y))
16.60/17.56	 SUM(U(x,y)) -> SUM(x)
16.60/17.56	 SUM(U(x,y)) -> SUM(y)
16.60/17.56	 SUM(empty) -> 0#(#)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.56	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	 U#(U(x4,x5),x6) -> U#(x4,x5)
16.60/17.56	 U#(x4,U(x5,x6)) -> U#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 U#(U(empty,b),x4) -> U#(b,x4)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 U#(U(x4,x5),x6) -> U#(x4,U(x5,x6))
16.60/17.56	 U#(x4,x5) -> U#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 U#(U(x4,x5),x6) -> U#(x4,x5)
16.60/17.56	 U#(x4,U(x5,x6)) -> U#(x5,x6)
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(0(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 SUM(U(x,y)) -> SUM(x)
16.60/17.56	 SUM(U(x,y)) -> SUM(y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 Empty
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(x,y)
16.60/17.56	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.56	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.56	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.56	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.56	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.56	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.56	 *#(+(y,z),x) -> *#(x,y)
16.60/17.56	 *#(+(y,z),x) -> *#(x,z)
16.60/17.56	 *#(0(x),y) -> *#(x,y)
16.60/17.56	 *#(1(x),y) -> *#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.56	 *#(x4,x5) -> *#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.56	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 PROD(U(x,y)) -> PROD(x)
16.60/17.56	 PROD(U(x,y)) -> PROD(y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 Empty
16.60/17.56	
16.60/17.56	
16.60/17.56	The problem is decomposed in 5 subproblems.
16.60/17.56	
16.60/17.56	Problem 1.1: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 U#(U(x4,x5),x6) = U#(x4,U(x5,x6))
16.60/17.56	 U#(x4,x5) = U#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 U#(U(empty,b),x4) -> U#(b,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 U(empty,b) -> b
16.60/17.56	-> SRules:
16.60/17.56	 U#(U(x4,x5),x6) -> U#(x4,x5)
16.60/17.56	 U#(x4,U(x5,x6)) -> U#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = 0
16.60/17.56	[0](X) = 0
16.60/17.56	[U](X1,X2) = X1 + X2
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 0
16.60/17.56	[1](X) = 0
16.60/17.56	[empty] = 2
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = 0
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 2.X1 + 2.X2
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.1: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 U#(U(x4,x5),x6) = U#(x4,U(x5,x6))
16.60/17.56	 U#(x4,x5) = U#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 Empty
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 U#(U(x4,x5),x6) -> U#(x4,x5)
16.60/17.56	 U#(x4,U(x5,x6)) -> U#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	 There is no strongly connected component
16.60/17.56	
16.60/17.56	The problem is finite.
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(0(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2 + 1
16.60/17.56	[0](X) = X + 1
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 0
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),0(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2
16.60/17.56	[0](X) = X + 1
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 1
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = X1 + X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(1(+(x,y)),x4)
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2
16.60/17.56	[0](X) = X + 1
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 1
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = X1 + X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(0(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2 + 2
16.60/17.56	[0](X) = X
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 0
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(#,x),x4) -> +#(x,x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2 + 1
16.60/17.56	[0](X) = X
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 1
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(0(+(1(#),+(x,y))),x4)
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2
16.60/17.56	[0](X) = X
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 1
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = X1 + X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(1(#),+(x,y))
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> Usable Rules:
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Interpretation type:
16.60/17.56	 Linear
16.60/17.56	->Coefficients:
16.60/17.56	 Natural Numbers
16.60/17.56	->Dimension:
16.60/17.56	 1
16.60/17.56	->Bound:
16.60/17.56	 2
16.60/17.56	->Interpretation:
16.60/17.56	 
16.60/17.56	[*](X1,X2) = 0
16.60/17.56	[+](X1,X2) = X1 + X2 + 1
16.60/17.56	[0](X) = X
16.60/17.56	[U](X1,X2) = 0
16.60/17.56	[prod](X) = 0
16.60/17.56	[sum](X) = 0
16.60/17.56	[#] = 1
16.60/17.56	[1](X) = X + 2
16.60/17.56	[empty] = 0
16.60/17.56	[singl](X) = 0
16.60/17.56	[*#](X1,X2) = 0
16.60/17.56	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.56	[0#](X) = 0
16.60/17.56	[U#](X1,X2) = 0
16.60/17.56	[PROD](X) = 0
16.60/17.56	[SUM](X) = 0
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	SCC Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	->Strongly Connected Components:
16.60/17.56	->->Cycle:
16.60/17.56	->->-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> FAxioms:
16.60/17.56	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) -> *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) -> +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) -> U(x5,x4)
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) -> +#(x5,x4)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	->->-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.56	 0(#) -> #
16.60/17.56	 U(empty,b) -> b
16.60/17.56	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.56	 prod(empty) -> 1(#)
16.60/17.56	 prod(singl(x)) -> x
16.60/17.56	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.56	 sum(empty) -> 0(#)
16.60/17.56	 sum(singl(x)) -> x
16.60/17.56	-> SRules:
16.60/17.56	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.56	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.56	
16.60/17.56	Problem 1.2: 
16.60/17.56	
16.60/17.56	Reduction Pairs Processor:
16.60/17.56	-> FAxioms:
16.60/17.56	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.56	 +#(x4,x5) = +#(x5,x4)
16.60/17.56	-> Pairs:
16.60/17.56	 +#(+(1(x),1(y)),x4) -> +#(x,y)
16.60/17.56	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.56	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.56	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.56	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.56	-> EAxioms:
16.60/17.56	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.56	 *(x4,x5) = *(x5,x4)
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.56	 U(x4,x5) = U(x5,x4)
16.60/17.56	-> Usable Equations:
16.60/17.56	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.56	 +(x4,x5) = +(x5,x4)
16.60/17.56	-> Rules:
16.60/17.56	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.56	 *(0(x),y) -> 0(*(x,y))
16.60/17.56	 *(#,x) -> #
16.60/17.56	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.56	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.56	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.56	 +(#,x) -> x
16.60/17.56	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Linear
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 2
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 0
16.60/17.57	[+](X1,X2) = X1 + X2
16.60/17.57	[0](X) = 2.X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = 2.X + 2
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 0
16.60/17.57	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) -> +#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,x5)
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Linear
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 2
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 0
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 0
16.60/17.57	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) -> +#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(0(x),0(y)) -> +#(x,y)
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Linear
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 2
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 0
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X + 1
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 2
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 0
16.60/17.57	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) -> +#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(0(x),1(y)) -> +#(x,y)
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Linear
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 2
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 0
16.60/17.57	[+](X1,X2) = X1 + X2 + 2
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 2
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 0
16.60/17.57	[+#](X1,X2) = X1 + X2
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) -> +#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(1(x),1(y)) -> +#(1(#),+(x,y))
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Linear
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 2
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 0
16.60/17.57	[+](X1,X2) = X1 + X2
16.60/17.57	[0](X) = X + 1
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 2
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 0
16.60/17.57	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 +#(+(x4,x5),x6) -> +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) -> +#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 +#(1(x),1(y)) -> +#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Linear
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 2
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 0
16.60/17.57	[+](X1,X2) = X1 + X2
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 1
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 0
16.60/17.57	[+#](X1,X2) = 2.X1 + 2.X2
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.2: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 +#(+(x4,x5),x6) = +#(x4,+(x5,x6))
16.60/17.57	 +#(x4,x5) = +#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 Empty
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 +#(x4,+(x5,x6)) -> +#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	 There is no strongly connected component
16.60/17.57	
16.60/17.57	The problem is finite.
16.60/17.57	
16.60/17.57	Problem 1.3: 
16.60/17.57	
16.60/17.57	Subterm Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 Empty
16.60/17.57	-> Pairs:
16.60/17.57	 SUM(U(x,y)) -> SUM(x)
16.60/17.57	 SUM(U(x,y)) -> SUM(y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 Empty
16.60/17.57	->Projection:
16.60/17.57	 pi(SUM) = [1]
16.60/17.57	
16.60/17.57	Problem 1.3: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 Empty
16.60/17.57	-> Pairs:
16.60/17.57	 Empty
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 Empty
16.60/17.57	->Strongly Connected Components:
16.60/17.57	 There is no strongly connected component
16.60/17.57	
16.60/17.57	The problem is finite.
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(x,y)
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 1
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(x,z)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 1
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 1
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(+(y,z),x) -> *#(x,y)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 1
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(+(y,z),x) -> *#(x,z)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 1
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	 *#(1(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 Natural Numbers
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 1
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 1
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(+(y,z),x),x4) -> *#(+(*(x,y),*(x,z)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Usable Equations:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> Usable Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Interpretation type:
16.60/17.57	 Simple mixed
16.60/17.57	->Coefficients:
16.60/17.57	 All rationals
16.60/17.57	->Dimension:
16.60/17.57	 1
16.60/17.57	->Bound:
16.60/17.57	 3
16.60/17.57	->Interpretation:
16.60/17.57	 
16.60/17.57	[*](X1,X2) = 3.X1.X2 + 2.X1 + 2.X2 + 2/3
16.60/17.57	[+](X1,X2) = X1 + X2 + 1
16.60/17.57	[0](X) = X + 1/2
16.60/17.57	[U](X1,X2) = 0
16.60/17.57	[prod](X) = 0
16.60/17.57	[sum](X) = 0
16.60/17.57	[#] = 0
16.60/17.57	[1](X) = X + 3/2
16.60/17.57	[empty] = 0
16.60/17.57	[singl](X) = 0
16.60/17.57	[*#](X1,X2) = 1/2.X1.X2 + 1/3.X1 + 1/3.X2
16.60/17.57	[+#](X1,X2) = 0
16.60/17.57	[0#](X) = 0
16.60/17.57	[U#](X1,X2) = 0
16.60/17.57	[PROD](X) = 0
16.60/17.57	[SUM](X) = 0
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	SCC Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	->Strongly Connected Components:
16.60/17.57	->->Cycle:
16.60/17.57	->->-> Pairs:
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	-> FAxioms:
16.60/17.57	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) -> *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) -> +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) -> U(x5,x4)
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) -> *#(x5,x4)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.57	 U(x4,x5) = U(x5,x4)
16.60/17.57	->->-> Rules:
16.60/17.57	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.57	 *(0(x),y) -> 0(*(x,y))
16.60/17.57	 *(#,x) -> #
16.60/17.57	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.57	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.57	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.57	 +(#,x) -> x
16.60/17.57	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.57	 0(#) -> #
16.60/17.57	 U(empty,b) -> b
16.60/17.57	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.57	 prod(empty) -> 1(#)
16.60/17.57	 prod(singl(x)) -> x
16.60/17.57	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.57	 sum(empty) -> 0(#)
16.60/17.57	 sum(singl(x)) -> x
16.60/17.57	-> SRules:
16.60/17.57	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.57	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.57	
16.60/17.57	Problem 1.4: 
16.60/17.57	
16.60/17.57	Reduction Pairs Processor:
16.60/17.57	-> FAxioms:
16.60/17.57	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.57	 *#(x4,x5) = *#(x5,x4)
16.60/17.57	-> Pairs:
16.60/17.57	 *#(*(0(x),y),x4) -> *#(0(*(x,y)),x4)
16.60/17.57	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.57	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.57	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.57	 *#(0(x),y) -> *#(x,y)
16.60/17.57	-> EAxioms:
16.60/17.57	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.57	 *(x4,x5) = *(x5,x4)
16.60/17.57	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.57	 +(x4,x5) = +(x5,x4)
16.60/17.57	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Usable Equations:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> Usable Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Interpretation type:
16.60/17.58	 Simple mixed
16.60/17.58	->Coefficients:
16.60/17.58	 All rationals
16.60/17.58	->Dimension:
16.60/17.58	 1
16.60/17.58	->Bound:
16.60/17.58	 3
16.60/17.58	->Interpretation:
16.60/17.58	 
16.60/17.58	[*](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1
16.60/17.58	[+](X1,X2) = X1 + X2 + 1
16.60/17.58	[0](X) = X + 1
16.60/17.58	[U](X1,X2) = 0
16.60/17.58	[prod](X) = 0
16.60/17.58	[sum](X) = 0
16.60/17.58	[#] = 1/3
16.60/17.58	[1](X) = X + 3
16.60/17.58	[empty] = 0
16.60/17.58	[singl](X) = 0
16.60/17.58	[*#](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2
16.60/17.58	[+#](X1,X2) = 0
16.60/17.58	[0#](X) = 0
16.60/17.58	[U#](X1,X2) = 0
16.60/17.58	[PROD](X) = 0
16.60/17.58	[SUM](X) = 0
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.58	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Strongly Connected Components:
16.60/17.58	->->Cycle:
16.60/17.58	->->-> Pairs:
16.60/17.58	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.58	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> FAxioms:
16.60/17.58	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) -> *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) -> +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) -> U(x5,x4)
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) -> *#(x5,x4)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	->->-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	Reduction Pairs Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(*(0(x),y),x4) -> *#(x,y)
16.60/17.58	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Usable Equations:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> Usable Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Interpretation type:
16.60/17.58	 Simple mixed
16.60/17.58	->Coefficients:
16.60/17.58	 All rationals
16.60/17.58	->Dimension:
16.60/17.58	 1
16.60/17.58	->Bound:
16.60/17.58	 3
16.60/17.58	->Interpretation:
16.60/17.58	 
16.60/17.58	[*](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
16.60/17.58	[+](X1,X2) = X1 + X2 + 1
16.60/17.58	[0](X) = X
16.60/17.58	[U](X1,X2) = 0
16.60/17.58	[prod](X) = 0
16.60/17.58	[sum](X) = 0
16.60/17.58	[#] = 0
16.60/17.58	[1](X) = X + 1
16.60/17.58	[empty] = 0
16.60/17.58	[singl](X) = 0
16.60/17.58	[*#](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2
16.60/17.58	[+#](X1,X2) = 0
16.60/17.58	[0#](X) = 0
16.60/17.58	[U#](X1,X2) = 0
16.60/17.58	[PROD](X) = 0
16.60/17.58	[SUM](X) = 0
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Strongly Connected Components:
16.60/17.58	->->Cycle:
16.60/17.58	->->-> Pairs:
16.60/17.58	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> FAxioms:
16.60/17.58	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) -> *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) -> +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) -> U(x5,x4)
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) -> *#(x5,x4)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	->->-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	Reduction Pairs Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(*(#,x),x4) -> *#(#,x4)
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Usable Equations:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> Usable Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Interpretation type:
16.60/17.58	 Simple mixed
16.60/17.58	->Coefficients:
16.60/17.58	 All rationals
16.60/17.58	->Dimension:
16.60/17.58	 1
16.60/17.58	->Bound:
16.60/17.58	 3
16.60/17.58	->Interpretation:
16.60/17.58	 
16.60/17.58	[*](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3
16.60/17.58	[+](X1,X2) = X1 + X2 + 3/2
16.60/17.58	[0](X) = X
16.60/17.58	[U](X1,X2) = 0
16.60/17.58	[prod](X) = 0
16.60/17.58	[sum](X) = 0
16.60/17.58	[#] = 1/2
16.60/17.58	[1](X) = X + 3
16.60/17.58	[empty] = 0
16.60/17.58	[singl](X) = 0
16.60/17.58	[*#](X1,X2) = 1/3.X1.X2 + 1/2.X1 + 1/2.X2
16.60/17.58	[+#](X1,X2) = 0
16.60/17.58	[0#](X) = 0
16.60/17.58	[U#](X1,X2) = 0
16.60/17.58	[PROD](X) = 0
16.60/17.58	[SUM](X) = 0
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Strongly Connected Components:
16.60/17.58	->->Cycle:
16.60/17.58	->->-> Pairs:
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> FAxioms:
16.60/17.58	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) -> *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) -> +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) -> U(x5,x4)
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) -> *#(x5,x4)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	->->-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	Reduction Pairs Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(*(1(x),y),x4) -> *#(+(0(*(x,y)),y),x4)
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Usable Equations:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> Usable Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Interpretation type:
16.60/17.58	 Simple mixed
16.60/17.58	->Coefficients:
16.60/17.58	 All rationals
16.60/17.58	->Dimension:
16.60/17.58	 1
16.60/17.58	->Bound:
16.60/17.58	 3
16.60/17.58	->Interpretation:
16.60/17.58	 
16.60/17.58	[*](X1,X2) = 3.X1.X2 + X1 + X2
16.60/17.58	[+](X1,X2) = X1 + X2 + 1/3
16.60/17.58	[0](X) = 3/2.X + 2/3
16.60/17.58	[U](X1,X2) = 0
16.60/17.58	[prod](X) = 0
16.60/17.58	[sum](X) = 0
16.60/17.58	[#] = 0
16.60/17.58	[1](X) = 3/2.X + 3
16.60/17.58	[empty] = 0
16.60/17.58	[singl](X) = 0
16.60/17.58	[*#](X1,X2) = X1.X2 + 1/3.X1 + 1/3.X2
16.60/17.58	[+#](X1,X2) = 0
16.60/17.58	[0#](X) = 0
16.60/17.58	[U#](X1,X2) = 0
16.60/17.58	[PROD](X) = 0
16.60/17.58	[SUM](X) = 0
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Strongly Connected Components:
16.60/17.58	->->Cycle:
16.60/17.58	->->-> Pairs:
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> FAxioms:
16.60/17.58	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) -> *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) -> +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) -> U(x5,x4)
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) -> *#(x5,x4)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	->->-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	Reduction Pairs Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Usable Equations:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> Usable Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	-> SRules:
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,x5)
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Interpretation type:
16.60/17.58	 Simple mixed
16.60/17.58	->Coefficients:
16.60/17.58	 All rationals
16.60/17.58	->Dimension:
16.60/17.58	 1
16.60/17.58	->Bound:
16.60/17.58	 3
16.60/17.58	->Interpretation:
16.60/17.58	 
16.60/17.58	[*](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
16.60/17.58	[+](X1,X2) = X1 + X2 + 3
16.60/17.58	[0](X) = X
16.60/17.58	[U](X1,X2) = 0
16.60/17.58	[prod](X) = 0
16.60/17.58	[sum](X) = 0
16.60/17.58	[#] = 0
16.60/17.58	[1](X) = X + 3
16.60/17.58	[empty] = 0
16.60/17.58	[singl](X) = 0
16.60/17.58	[*#](X1,X2) = X1.X2 + X1 + X2
16.60/17.58	[+#](X1,X2) = 0
16.60/17.58	[0#](X) = 0
16.60/17.58	[U#](X1,X2) = 0
16.60/17.58	[PROD](X) = 0
16.60/17.58	[SUM](X) = 0
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Strongly Connected Components:
16.60/17.58	->->Cycle:
16.60/17.58	->->-> Pairs:
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> FAxioms:
16.60/17.58	 *(*(x4,x5),x6) -> *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) -> *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) -> +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) -> +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) -> U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) -> U(x5,x4)
16.60/17.58	 *#(*(x4,x5),x6) -> *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) -> *#(x5,x4)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	->->-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	Reduction Pairs Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 *#(0(x),y) -> *#(x,y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Usable Equations:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> Usable Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	-> SRules:
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Interpretation type:
16.60/17.58	 Simple mixed
16.60/17.58	->Coefficients:
16.60/17.58	 All rationals
16.60/17.58	->Dimension:
16.60/17.58	 1
16.60/17.58	->Bound:
16.60/17.58	 3
16.60/17.58	->Interpretation:
16.60/17.58	 
16.60/17.58	[*](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
16.60/17.58	[+](X1,X2) = X1 + X2 + 1
16.60/17.58	[0](X) = X + 2/3
16.60/17.58	[U](X1,X2) = 0
16.60/17.58	[prod](X) = 0
16.60/17.58	[sum](X) = 0
16.60/17.58	[#] = 1/3
16.60/17.58	[1](X) = X + 2
16.60/17.58	[empty] = 0
16.60/17.58	[singl](X) = 0
16.60/17.58	[*#](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2
16.60/17.58	[+#](X1,X2) = 0
16.60/17.58	[0#](X) = 0
16.60/17.58	[U#](X1,X2) = 0
16.60/17.58	[PROD](X) = 0
16.60/17.58	[SUM](X) = 0
16.60/17.58	
16.60/17.58	Problem 1.4: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 *#(*(x4,x5),x6) = *#(x4,*(x5,x6))
16.60/17.58	 *#(x4,x5) = *#(x5,x4)
16.60/17.58	-> Pairs:
16.60/17.58	 Empty
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 *#(x4,*(x5,x6)) -> *#(x5,x6)
16.60/17.58	->Strongly Connected Components:
16.60/17.58	 There is no strongly connected component
16.60/17.58	
16.60/17.58	The problem is finite.
16.60/17.58	
16.60/17.58	Problem 1.5: 
16.60/17.58	
16.60/17.58	Subterm Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 Empty
16.60/17.58	-> Pairs:
16.60/17.58	 PROD(U(x,y)) -> PROD(x)
16.60/17.58	 PROD(U(x,y)) -> PROD(y)
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 Empty
16.60/17.58	->Projection:
16.60/17.58	 pi(PROD) = [1]
16.60/17.58	
16.60/17.58	Problem 1.5: 
16.60/17.58	
16.60/17.58	SCC Processor:
16.60/17.58	-> FAxioms:
16.60/17.58	 Empty
16.60/17.58	-> Pairs:
16.60/17.58	 Empty
16.60/17.58	-> EAxioms:
16.60/17.58	 *(*(x4,x5),x6) = *(x4,*(x5,x6))
16.60/17.58	 *(x4,x5) = *(x5,x4)
16.60/17.58	 +(+(x4,x5),x6) = +(x4,+(x5,x6))
16.60/17.58	 +(x4,x5) = +(x5,x4)
16.60/17.58	 U(U(x4,x5),x6) = U(x4,U(x5,x6))
16.60/17.58	 U(x4,x5) = U(x5,x4)
16.60/17.58	-> Rules:
16.60/17.58	 *(+(y,z),x) -> +(*(x,y),*(x,z))
16.60/17.58	 *(0(x),y) -> 0(*(x,y))
16.60/17.58	 *(#,x) -> #
16.60/17.58	 *(1(x),y) -> +(0(*(x,y)),y)
16.60/17.58	 +(0(x),0(y)) -> 0(+(x,y))
16.60/17.58	 +(0(x),1(y)) -> 1(+(x,y))
16.60/17.58	 +(#,x) -> x
16.60/17.58	 +(1(x),1(y)) -> 0(+(1(#),+(x,y)))
16.60/17.58	 0(#) -> #
16.60/17.58	 U(empty,b) -> b
16.60/17.58	 prod(U(x,y)) -> *(prod(x),prod(y))
16.60/17.58	 prod(empty) -> 1(#)
16.60/17.58	 prod(singl(x)) -> x
16.60/17.58	 sum(U(x,y)) -> +(sum(x),sum(y))
16.60/17.58	 sum(empty) -> 0(#)
16.60/17.58	 sum(singl(x)) -> x
16.60/17.58	-> SRules:
16.60/17.58	 Empty
16.60/17.58	->Strongly Connected Components:
16.60/17.58	 There is no strongly connected component
16.60/17.58	
16.60/17.58	The problem is finite.
16.60/17.58	EOF