16.09/16.84	YES
16.09/16.84	
16.09/16.84	Problem 1: 
16.09/16.84	
16.09/16.84	(VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S)
16.09/16.84	(RULES
16.09/16.84	active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	active(x(N:S,0)) -> mark(0)
16.09/16.84	and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	mark(0) -> active(0)
16.09/16.84	mark(tt) -> active(tt)
16.09/16.84	plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	s(active(X:S)) -> s(X:S)
16.09/16.84	s(mark(X:S)) -> s(X:S)
16.09/16.84	x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	) 
16.09/16.84	(STRATEGY INNERMOST)
16.09/16.84	
16.09/16.84	Problem 1: 
16.09/16.84	
16.09/16.84	Dependency Pairs Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> PLUS(N:S,M:S)
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> S(plus(N:S,M:S))
16.09/16.84	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> PLUS(x(N:S,M:S),N:S)
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> X(N:S,M:S)
16.09/16.84	 ACTIVE(x(N:S,0)) -> MARK(0)
16.09/16.84	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.84	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.84	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.84	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.84	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.84	 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
16.09/16.84	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
16.09/16.84	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.09/16.84	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.84	 MARK(s(X:S)) -> S(mark(X:S))
16.09/16.84	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.84	 MARK(x(X1:S,X2:S)) -> X(mark(X1:S),mark(X2:S))
16.09/16.84	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.84	 S(active(X:S)) -> S(X:S)
16.09/16.84	 S(mark(X:S)) -> S(X:S)
16.09/16.84	 X(active(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(mark(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	
16.09/16.84	Problem 1: 
16.09/16.84	
16.09/16.84	SCC Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> PLUS(N:S,M:S)
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> S(plus(N:S,M:S))
16.09/16.84	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> PLUS(x(N:S,M:S),N:S)
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> X(N:S,M:S)
16.09/16.84	 ACTIVE(x(N:S,0)) -> MARK(0)
16.09/16.84	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.84	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.84	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.84	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.84	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.84	 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
16.09/16.84	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
16.09/16.84	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.09/16.84	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.84	 MARK(s(X:S)) -> S(mark(X:S))
16.09/16.84	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.84	 MARK(x(X1:S,X2:S)) -> X(mark(X1:S),mark(X2:S))
16.09/16.84	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.84	 S(active(X:S)) -> S(X:S)
16.09/16.84	 S(mark(X:S)) -> S(X:S)
16.09/16.84	 X(active(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(mark(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->Strongly Connected Components:
16.09/16.84	->->Cycle:
16.09/16.84	->->-> Pairs:
16.09/16.84	 X(active(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(mark(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	->->-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->->Cycle:
16.09/16.84	->->-> Pairs:
16.09/16.84	 S(active(X:S)) -> S(X:S)
16.09/16.84	 S(mark(X:S)) -> S(X:S)
16.09/16.84	->->-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->->Cycle:
16.09/16.84	->->-> Pairs:
16.09/16.84	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.84	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.84	->->-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->->Cycle:
16.09/16.84	->->-> Pairs:
16.09/16.84	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.84	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.84	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.84	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.84	->->-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->->Cycle:
16.09/16.84	->->-> Pairs:
16.09/16.84	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.84	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.84	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.84	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.84	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.84	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.84	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.09/16.84	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.84	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.84	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.84	->->-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	
16.09/16.84	
16.09/16.84	The problem is decomposed in 5 subproblems.
16.09/16.84	
16.09/16.84	Problem 1.1: 
16.09/16.84	
16.09/16.84	Subterm Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 X(active(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(mark(X1:S),X2:S) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->Projection:
16.09/16.84	 pi(X) = 1
16.09/16.84	
16.09/16.84	Problem 1.1: 
16.09/16.84	
16.09/16.84	SCC Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->Strongly Connected Components:
16.09/16.84	->->Cycle:
16.09/16.84	->->-> Pairs:
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	->->-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	
16.09/16.84	Problem 1.1: 
16.09/16.84	
16.09/16.84	Subterm Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 X(X1:S,active(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	 X(X1:S,mark(X2:S)) -> X(X1:S,X2:S)
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->Projection:
16.09/16.84	 pi(X) = 2
16.09/16.84	
16.09/16.84	Problem 1.1: 
16.09/16.84	
16.09/16.84	SCC Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 Empty
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.84	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.84	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.84	 active(x(N:S,0)) -> mark(0)
16.09/16.84	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.84	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.84	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.84	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.84	 mark(0) -> active(0)
16.09/16.84	 mark(tt) -> active(tt)
16.09/16.84	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.84	 s(active(X:S)) -> s(X:S)
16.09/16.84	 s(mark(X:S)) -> s(X:S)
16.09/16.84	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.84	->Strongly Connected Components:
16.09/16.84	 There is no strongly connected component
16.09/16.84	
16.09/16.84	The problem is finite.
16.09/16.84	
16.09/16.84	Problem 1.2: 
16.09/16.84	
16.09/16.84	Subterm Processor:
16.09/16.84	-> Pairs:
16.09/16.84	 S(active(X:S)) -> S(X:S)
16.09/16.84	 S(mark(X:S)) -> S(X:S)
16.09/16.84	-> Rules:
16.09/16.84	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.84	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Projection:
16.09/16.85	 pi(S) = 1
16.09/16.85	
16.09/16.85	Problem 1.2: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 Empty
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	 There is no strongly connected component
16.09/16.85	
16.09/16.85	The problem is finite.
16.09/16.85	
16.09/16.85	Problem 1.3: 
16.09/16.85	
16.09/16.85	Subterm Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.85	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.09/16.85	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Projection:
16.09/16.85	 pi(PLUS) = 1
16.09/16.85	
16.09/16.85	Problem 1.3: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.3: 
16.09/16.85	
16.09/16.85	Subterm Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Projection:
16.09/16.85	 pi(PLUS) = 2
16.09/16.85	
16.09/16.85	Problem 1.3: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 Empty
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	 There is no strongly connected component
16.09/16.85	
16.09/16.85	The problem is finite.
16.09/16.85	
16.09/16.85	Problem 1.4: 
16.09/16.85	
16.09/16.85	Subterm Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.85	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.09/16.85	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Projection:
16.09/16.85	 pi(AND) = 1
16.09/16.85	
16.09/16.85	Problem 1.4: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.4: 
16.09/16.85	
16.09/16.85	Subterm Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Projection:
16.09/16.85	 pi(AND) = 2
16.09/16.85	
16.09/16.85	Problem 1.4: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 Empty
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	 There is no strongly connected component
16.09/16.85	
16.09/16.85	The problem is finite.
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Linear
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = 2
16.09/16.85	[and](X1,X2) = 1
16.09/16.85	[mark](X) = 2
16.09/16.85	[plus](X1,X2) = 1
16.09/16.85	[s](X) = 0
16.09/16.85	[x](X1,X2) = 1
16.09/16.85	[0] = 2
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 1
16.09/16.85	[ACTIVE](X) = 2.X
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Simple mixed
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = X
16.09/16.85	[and](X1,X2) = 2.X1 + 2.X2 + 2
16.09/16.85	[mark](X) = X
16.09/16.85	[plus](X1,X2) = X1 + 2.X2 + 2
16.09/16.85	[s](X) = X + 2
16.09/16.85	[x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2
16.09/16.85	[0] = 2
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 0
16.09/16.85	[ACTIVE](X) = 2.X.X + X + 2
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2.X.X + 2.X + 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Linear
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = 2
16.09/16.85	[and](X1,X2) = 0
16.09/16.85	[mark](X) = 2
16.09/16.85	[plus](X1,X2) = 1
16.09/16.85	[s](X) = 0
16.09/16.85	[x](X1,X2) = 1
16.09/16.85	[0] = 0
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 2
16.09/16.85	[ACTIVE](X) = 2.X
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S)))
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Simple mixed
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = X
16.09/16.85	[and](X1,X2) = X1.X2 + X1 + X2 + 2
16.09/16.85	[mark](X) = X
16.09/16.85	[plus](X1,X2) = X1 + 2.X2 + 2
16.09/16.85	[s](X) = X + 2
16.09/16.85	[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
16.09/16.85	[0] = 2
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 0
16.09/16.85	[ACTIVE](X) = 2.X.X + 2.X + 2
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2.X.X + 2.X + 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(plus(N:S,0)) -> MARK(N:S)
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Simple mixed
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = X
16.09/16.85	[and](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1
16.09/16.85	[mark](X) = X
16.09/16.85	[plus](X1,X2) = X1 + 2.X2 + 2
16.09/16.85	[s](X) = X + 2
16.09/16.85	[x](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 1
16.09/16.85	[0] = 0
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 2
16.09/16.85	[ACTIVE](X) = 2.X + 2
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2.X + 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Linear
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = 2
16.09/16.85	[and](X1,X2) = 2
16.09/16.85	[mark](X) = 2
16.09/16.85	[plus](X1,X2) = 0
16.09/16.85	[s](X) = 2
16.09/16.85	[x](X1,X2) = 1
16.09/16.85	[0] = 0
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 2
16.09/16.85	[ACTIVE](X) = X + 1
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Reduction Pairs Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S))
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	-> Usable rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Interpretation type:
16.09/16.85	 Simple mixed
16.09/16.85	->Coefficients:
16.09/16.85	 Natural Numbers
16.09/16.85	->Dimension:
16.09/16.85	 1
16.09/16.85	->Bound:
16.09/16.85	 2
16.09/16.85	->Interpretation:
16.09/16.85	 
16.09/16.85	[active](X) = X
16.09/16.85	[and](X1,X2) = 2.X1 + X2 + 2
16.09/16.85	[mark](X) = X
16.09/16.85	[plus](X1,X2) = X1 + 2.X2
16.09/16.85	[s](X) = X + 1
16.09/16.85	[x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2
16.09/16.85	[0] = 1
16.09/16.85	[fSNonEmpty] = 0
16.09/16.85	[tt] = 0
16.09/16.85	[ACTIVE](X) = 2.X.X + 2.X + 1
16.09/16.85	[AND](X1,X2) = 0
16.09/16.85	[MARK](X) = 2.X.X + 2.X + 2
16.09/16.85	[PLUS](X1,X2) = 0
16.09/16.85	[S](X) = 0
16.09/16.85	[X](X1,X2) = 0
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	->->Cycle:
16.09/16.85	->->-> Pairs:
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	->->-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	Subterm Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	 MARK(s(X:S)) -> MARK(X:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X1:S)
16.09/16.85	 MARK(x(X1:S,X2:S)) -> MARK(X2:S)
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Projection:
16.09/16.85	 pi(MARK) = 1
16.09/16.85	
16.09/16.85	Problem 1.5: 
16.09/16.85	
16.09/16.85	SCC Processor:
16.09/16.85	-> Pairs:
16.09/16.85	 Empty
16.09/16.85	-> Rules:
16.09/16.85	 active(and(tt,X:S)) -> mark(X:S)
16.09/16.85	 active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S)))
16.09/16.85	 active(plus(N:S,0)) -> mark(N:S)
16.09/16.85	 active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S))
16.09/16.85	 active(x(N:S,0)) -> mark(0)
16.09/16.85	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.09/16.85	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.09/16.85	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(s(X:S)) -> active(s(mark(X:S)))
16.09/16.85	 mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S)))
16.09/16.85	 mark(0) -> active(0)
16.09/16.85	 mark(tt) -> active(tt)
16.09/16.85	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.09/16.85	 s(active(X:S)) -> s(X:S)
16.09/16.85	 s(mark(X:S)) -> s(X:S)
16.09/16.85	 x(active(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(mark(X1:S),X2:S) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,active(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	 x(X1:S,mark(X2:S)) -> x(X1:S,X2:S)
16.09/16.85	->Strongly Connected Components:
16.09/16.85	 There is no strongly connected component
16.09/16.85	
16.09/16.85	The problem is finite.
16.09/16.85	EOF