6.72/7.86	YES
6.72/7.86	
6.72/7.86	Problem 1: 
6.72/7.86	
6.72/7.86	(VAR x y)
6.72/7.86	(THEORY 
6.72/7.86	(AC plus times))
6.72/7.86	(RULES
6.72/7.86	0(S) -> S
6.72/7.86	minus(x,y) -> plus(opp(y),x)
6.72/7.86	opp(0(x)) -> 0(opp(x))
6.72/7.86	opp(1(x)) -> j(opp(x))
6.72/7.86	opp(S) -> S
6.72/7.86	opp(j(x)) -> 1(opp(x))
6.72/7.86	plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	plus(S,x) -> x
6.72/7.86	plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	times(0(x),y) -> 0(times(x,y))
6.72/7.86	times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	times(S,x) -> S
6.72/7.86	times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	)
6.72/7.86	
6.72/7.86	Problem 1: 
6.72/7.86	
6.72/7.86	Dependency Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.86	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 MINUS(x,y) -> OPP(y)
6.72/7.86	 MINUS(x,y) -> PLUS(opp(y),x)
6.72/7.86	 OPP(0(x)) -> 0#(opp(x))
6.72/7.86	 OPP(0(x)) -> OPP(x)
6.72/7.86	 OPP(1(x)) -> OPP(x)
6.72/7.86	 OPP(j(x)) -> OPP(x)
6.72/7.86	 PLUS(0(x),0(y)) -> 0#(plus(x,y))
6.72/7.86	 PLUS(0(x),0(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y))
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> 0#(plus(x,y))
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> 0#(plus(x,y))
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	 TIMES(0(x),y) -> 0#(times(x,y))
6.72/7.86	 TIMES(0(x),y) -> TIMES(x,y)
6.72/7.86	 TIMES(times(0(x),y),x2) -> 0#(times(x,y))
6.72/7.86	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.86	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(times(1(x),y),x2) -> 0#(times(x,y))
6.72/7.86	 TIMES(times(1(x),y),x2) -> PLUS(0(times(x,y)),y)
6.72/7.86	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.86	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.86	 TIMES(times(j(x),y),x2) -> 0#(times(x,y))
6.72/7.86	 TIMES(times(j(x),y),x2) -> MINUS(0(times(x,y)),y)
6.72/7.86	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.86	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(1(x),y) -> 0#(times(x,y))
6.72/7.86	 TIMES(1(x),y) -> PLUS(0(times(x,y)),y)
6.72/7.86	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.86	 TIMES(j(x),y) -> 0#(times(x,y))
6.72/7.86	 TIMES(j(x),y) -> MINUS(0(times(x,y)),y)
6.72/7.86	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.86	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.86	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 MINUS(x,y) -> OPP(y)
6.72/7.86	 MINUS(x,y) -> PLUS(opp(y),x)
6.72/7.86	 OPP(0(x)) -> 0#(opp(x))
6.72/7.86	 OPP(0(x)) -> OPP(x)
6.72/7.86	 OPP(1(x)) -> OPP(x)
6.72/7.86	 OPP(j(x)) -> OPP(x)
6.72/7.86	 PLUS(0(x),0(y)) -> 0#(plus(x,y))
6.72/7.86	 PLUS(0(x),0(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y))
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> 0#(plus(x,y))
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> 0#(plus(x,y))
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	 TIMES(0(x),y) -> 0#(times(x,y))
6.72/7.86	 TIMES(0(x),y) -> TIMES(x,y)
6.72/7.86	 TIMES(times(0(x),y),x2) -> 0#(times(x,y))
6.72/7.86	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.86	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(times(1(x),y),x2) -> 0#(times(x,y))
6.72/7.86	 TIMES(times(1(x),y),x2) -> PLUS(0(times(x,y)),y)
6.72/7.86	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.86	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.86	 TIMES(times(j(x),y),x2) -> 0#(times(x,y))
6.72/7.86	 TIMES(times(j(x),y),x2) -> MINUS(0(times(x,y)),y)
6.72/7.86	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.86	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(1(x),y) -> 0#(times(x,y))
6.72/7.86	 TIMES(1(x),y) -> PLUS(0(times(x,y)),y)
6.72/7.86	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.86	 TIMES(j(x),y) -> 0#(times(x,y))
6.72/7.86	 TIMES(j(x),y) -> MINUS(0(times(x,y)),y)
6.72/7.86	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.86	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(0(x),0(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 OPP(0(x)) -> OPP(x)
6.72/7.86	 OPP(1(x)) -> OPP(x)
6.72/7.86	 OPP(j(x)) -> OPP(x)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 Empty
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 TIMES(0(x),y) -> TIMES(x,y)
6.72/7.86	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.86	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.86	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.86	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.86	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.86	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.86	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.86	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.86	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.86	
6.72/7.86	
6.72/7.86	The problem is decomposed in 3 subproblems.
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(0(x),0(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 2
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(0(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 2
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(0(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 1
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 2
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 2
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 1
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X + 1
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Strongly Connected Components:
6.72/7.86	->->Cycle:
6.72/7.86	->->-> Pairs:
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> FAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) -> plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) -> times(x3,x2)
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	->->-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	Reduction Pairs Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.86	-> EAxioms:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.86	 times(x2,x3) = times(x3,x2)
6.72/7.86	-> Usable Equations:
6.72/7.86	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.86	 plus(x2,x3) = plus(x3,x2)
6.72/7.86	-> Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 minus(x,y) -> plus(opp(y),x)
6.72/7.86	 opp(0(x)) -> 0(opp(x))
6.72/7.86	 opp(1(x)) -> j(opp(x))
6.72/7.86	 opp(S) -> S
6.72/7.86	 opp(j(x)) -> 1(opp(x))
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	 times(0(x),y) -> 0(times(x,y))
6.72/7.86	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.86	 times(S,x) -> S
6.72/7.86	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.86	-> Usable Rules:
6.72/7.86	 0(S) -> S
6.72/7.86	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.86	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.86	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.86	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.86	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.86	 plus(S,x) -> x
6.72/7.86	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.86	-> SRules:
6.72/7.86	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.86	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.86	->Interpretation type:
6.72/7.86	 Linear
6.72/7.86	->Coefficients:
6.72/7.86	 Natural Numbers
6.72/7.86	->Dimension:
6.72/7.86	 1
6.72/7.86	->Bound:
6.72/7.86	 2
6.72/7.86	->Interpretation:
6.72/7.86	 
6.72/7.86	[0](X) = X
6.72/7.86	[minus](X1,X2) = 0
6.72/7.86	[opp](X) = 0
6.72/7.86	[plus](X1,X2) = X1 + X2
6.72/7.86	[times](X1,X2) = 0
6.72/7.86	[1](X) = X + 2
6.72/7.86	[S] = 0
6.72/7.86	[j](X) = X + 2
6.72/7.86	[0#](X) = 0
6.72/7.86	[MINUS](X1,X2) = 0
6.72/7.86	[OPP](X) = 0
6.72/7.86	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.86	[TIMES](X1,X2) = 0
6.72/7.86	
6.72/7.86	Problem 1.1: 
6.72/7.86	
6.72/7.86	SCC Processor:
6.72/7.86	-> FAxioms:
6.72/7.86	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.86	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.86	-> Pairs:
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.86	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.86	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.86	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 1
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 1
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),1(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 1
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(1(x),j(y)) -> PLUS(x,y)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	 PLUS(j(x),j(y)) -> PLUS(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 1
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = X + 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 2
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2 + 2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = 2
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = 0
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(S,x),x2) -> PLUS(x,x2)
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = 0
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2 + 2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = 0
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = 1
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = X1 + X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) -> PLUS(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Linear
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 2
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = 2
6.72/7.87	[minus](X1,X2) = 0
6.72/7.87	[opp](X) = 0
6.72/7.87	[plus](X1,X2) = X1 + X2 + 2
6.72/7.87	[times](X1,X2) = 0
6.72/7.87	[1](X) = 1
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = 0
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 2.X1 + 2.X2
6.72/7.87	[TIMES](X1,X2) = 0
6.72/7.87	
6.72/7.87	Problem 1.1: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
6.72/7.87	 PLUS(x2,x3) = PLUS(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 Empty
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
6.72/7.87	 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	 There is no strongly connected component
6.72/7.87	
6.72/7.87	The problem is finite.
6.72/7.87	
6.72/7.87	Problem 1.2: 
6.72/7.87	
6.72/7.87	Subterm Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 Empty
6.72/7.87	-> Pairs:
6.72/7.87	 OPP(0(x)) -> OPP(x)
6.72/7.87	 OPP(1(x)) -> OPP(x)
6.72/7.87	 OPP(j(x)) -> OPP(x)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 Empty
6.72/7.87	->Projection:
6.72/7.87	 pi(OPP) = [1]
6.72/7.87	
6.72/7.87	Problem 1.2: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 Empty
6.72/7.87	-> Pairs:
6.72/7.87	 Empty
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 Empty
6.72/7.87	->Strongly Connected Components:
6.72/7.87	 There is no strongly connected component
6.72/7.87	
6.72/7.87	The problem is finite.
6.72/7.87	
6.72/7.87	Problem 1.3: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 TIMES(0(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Simple mixed
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 1
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 1
6.72/7.87	[minus](X1,X2) = X1 + X2
6.72/7.87	[opp](X) = X
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.87	[1](X) = X + 1
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 1
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 0
6.72/7.87	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.87	
6.72/7.87	Problem 1.3: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.3: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Simple mixed
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 1
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X + 1
6.72/7.87	[minus](X1,X2) = X1 + X2
6.72/7.87	[opp](X) = X
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.87	[1](X) = X + 1
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 1
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 0
6.72/7.87	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.87	
6.72/7.87	Problem 1.3: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	->Strongly Connected Components:
6.72/7.87	->->Cycle:
6.72/7.87	->->-> Pairs:
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> FAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) -> plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) -> times(x3,x2)
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	->->-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	
6.72/7.87	Problem 1.3: 
6.72/7.87	
6.72/7.87	Reduction Pairs Processor:
6.72/7.87	-> FAxioms:
6.72/7.87	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.87	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.87	-> Pairs:
6.72/7.87	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.87	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.87	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.87	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.87	-> EAxioms:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Usable Equations:
6.72/7.87	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.87	 plus(x2,x3) = plus(x3,x2)
6.72/7.87	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.87	 times(x2,x3) = times(x3,x2)
6.72/7.87	-> Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> Usable Rules:
6.72/7.87	 0(S) -> S
6.72/7.87	 minus(x,y) -> plus(opp(y),x)
6.72/7.87	 opp(0(x)) -> 0(opp(x))
6.72/7.87	 opp(1(x)) -> j(opp(x))
6.72/7.87	 opp(S) -> S
6.72/7.87	 opp(j(x)) -> 1(opp(x))
6.72/7.87	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.87	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.87	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.87	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.87	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.87	 plus(S,x) -> x
6.72/7.87	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.87	 times(0(x),y) -> 0(times(x,y))
6.72/7.87	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.87	 times(S,x) -> S
6.72/7.87	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.87	-> SRules:
6.72/7.87	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.87	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.87	->Interpretation type:
6.72/7.87	 Simple mixed
6.72/7.87	->Coefficients:
6.72/7.87	 Natural Numbers
6.72/7.87	->Dimension:
6.72/7.87	 1
6.72/7.87	->Bound:
6.72/7.87	 1
6.72/7.87	->Interpretation:
6.72/7.87	 
6.72/7.87	[0](X) = X
6.72/7.87	[minus](X1,X2) = X1 + X2 + 1
6.72/7.87	[opp](X) = X + 1
6.72/7.87	[plus](X1,X2) = X1 + X2
6.72/7.87	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.87	[1](X) = X + 1
6.72/7.87	[S] = 0
6.72/7.87	[j](X) = X + 1
6.72/7.87	[0#](X) = 0
6.72/7.87	[MINUS](X1,X2) = 0
6.72/7.87	[OPP](X) = 0
6.72/7.87	[PLUS](X1,X2) = 0
6.72/7.87	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.87	
6.72/7.87	Problem 1.3: 
6.72/7.87	
6.72/7.87	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(1(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Simple mixed
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 1
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = X + 1
6.72/7.88	[minus](X1,X2) = X1 + X2
6.72/7.88	[opp](X) = X
6.72/7.88	[plus](X1,X2) = X1 + X2
6.72/7.88	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	[1](X) = X + 1
6.72/7.88	[S] = 0
6.72/7.88	[j](X) = X + 1
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Simple mixed
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 1
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = X
6.72/7.88	[minus](X1,X2) = X1 + X2
6.72/7.88	[opp](X) = X
6.72/7.88	[plus](X1,X2) = X1 + X2
6.72/7.88	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	[1](X) = X + 1
6.72/7.88	[S] = 0
6.72/7.88	[j](X) = X + 1
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(times(j(x),y),x2) -> TIMES(x,y)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Simple mixed
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 1
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = X
6.72/7.88	[minus](X1,X2) = X1 + X2 + 1
6.72/7.88	[opp](X) = X + 1
6.72/7.88	[plus](X1,X2) = X1 + X2
6.72/7.88	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	[1](X) = X + 1
6.72/7.88	[S] = 0
6.72/7.88	[j](X) = X + 1
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(1(x),y) -> TIMES(x,y)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Simple mixed
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 1
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = X + 1
6.72/7.88	[minus](X1,X2) = X1 + X2
6.72/7.88	[opp](X) = X
6.72/7.88	[plus](X1,X2) = X1 + X2
6.72/7.88	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	[1](X) = X + 1
6.72/7.88	[S] = 0
6.72/7.88	[j](X) = X + 1
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	 TIMES(j(x),y) -> TIMES(x,y)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Simple mixed
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 1
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = X + 1
6.72/7.88	[minus](X1,X2) = X1 + X2
6.72/7.88	[opp](X) = X
6.72/7.88	[plus](X1,X2) = X1 + X2
6.72/7.88	[times](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	[1](X) = X + 1
6.72/7.88	[S] = 0
6.72/7.88	[j](X) = X + 1
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = X1.X2 + X1 + X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Linear
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 2
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = 1
6.72/7.88	[minus](X1,X2) = 2.X1 + X2 + 1
6.72/7.88	[opp](X) = X
6.72/7.88	[plus](X1,X2) = X1 + X2 + 1
6.72/7.88	[times](X1,X2) = X1 + X2 + 2
6.72/7.88	[1](X) = 1
6.72/7.88	[S] = 1
6.72/7.88	[j](X) = 1
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = 2.X1 + 2.X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	->->Cycle:
6.72/7.88	->->-> Pairs:
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	-> FAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) -> plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) -> times(x3,x2)
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) -> TIMES(x3,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	->->-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	Reduction Pairs Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 TIMES(times(S,x),x2) -> TIMES(S,x2)
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Usable Equations:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> Usable Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Interpretation type:
6.72/7.88	 Linear
6.72/7.88	->Coefficients:
6.72/7.88	 Natural Numbers
6.72/7.88	->Dimension:
6.72/7.88	 1
6.72/7.88	->Bound:
6.72/7.88	 2
6.72/7.88	->Interpretation:
6.72/7.88	 
6.72/7.88	[0](X) = 1
6.72/7.88	[minus](X1,X2) = X1 + X2 + 2
6.72/7.88	[opp](X) = X + 1
6.72/7.88	[plus](X1,X2) = X1 + X2 + 1
6.72/7.88	[times](X1,X2) = X1 + X2 + 1
6.72/7.88	[1](X) = 1
6.72/7.88	[S] = 1
6.72/7.88	[j](X) = 2
6.72/7.88	[0#](X) = 0
6.72/7.88	[MINUS](X1,X2) = 0
6.72/7.88	[OPP](X) = 0
6.72/7.88	[PLUS](X1,X2) = 0
6.72/7.88	[TIMES](X1,X2) = 2.X1 + 2.X2
6.72/7.88	
6.72/7.88	Problem 1.3: 
6.72/7.88	
6.72/7.88	SCC Processor:
6.72/7.88	-> FAxioms:
6.72/7.88	 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
6.72/7.88	 TIMES(x2,x3) = TIMES(x3,x2)
6.72/7.88	-> Pairs:
6.72/7.88	 Empty
6.72/7.88	-> EAxioms:
6.72/7.88	 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
6.72/7.88	 plus(x2,x3) = plus(x3,x2)
6.72/7.88	 times(times(x2,x3),x4) = times(x2,times(x3,x4))
6.72/7.88	 times(x2,x3) = times(x3,x2)
6.72/7.88	-> Rules:
6.72/7.88	 0(S) -> S
6.72/7.88	 minus(x,y) -> plus(opp(y),x)
6.72/7.88	 opp(0(x)) -> 0(opp(x))
6.72/7.88	 opp(1(x)) -> j(opp(x))
6.72/7.88	 opp(S) -> S
6.72/7.88	 opp(j(x)) -> 1(opp(x))
6.72/7.88	 plus(0(x),0(y)) -> 0(plus(x,y))
6.72/7.88	 plus(0(x),1(y)) -> 1(plus(x,y))
6.72/7.88	 plus(0(x),j(y)) -> j(plus(x,y))
6.72/7.88	 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
6.72/7.88	 plus(1(x),j(y)) -> 0(plus(x,y))
6.72/7.88	 plus(S,x) -> x
6.72/7.88	 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
6.72/7.88	 times(0(x),y) -> 0(times(x,y))
6.72/7.88	 times(1(x),y) -> plus(0(times(x,y)),y)
6.72/7.88	 times(S,x) -> S
6.72/7.88	 times(j(x),y) -> minus(0(times(x,y)),y)
6.72/7.88	-> SRules:
6.72/7.88	 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
6.72/7.88	 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
6.72/7.88	->Strongly Connected Components:
6.72/7.88	 There is no strongly connected component
6.72/7.88	
6.72/7.88	The problem is finite.
6.72/7.88	EOF