0.00/0.16	YES
0.00/0.16	
0.00/0.16	Problem 1: 
0.00/0.16	
0.00/0.16	(VAR v_NonEmpty:S X:S X1:S X2:S XS:S)
0.00/0.16	(RULES
0.00/0.16	active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	active(zeros) -> mark(cons(0,zeros))
0.00/0.16	cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	proper(0) -> ok(0)
0.00/0.16	proper(zeros) -> ok(zeros)
0.00/0.16	tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	top(ok(X:S)) -> top(active(X:S))
0.00/0.16	) 
0.00/0.16	(STRATEGY INNERMOST)
0.00/0.16	
0.00/0.16	Problem 1: 
0.00/0.16	
0.00/0.16	Dependency Pairs Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
0.00/0.16	 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
0.00/0.16	 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
0.00/0.16	 ACTIVE(tail(X:S)) -> TAIL(active(X:S))
0.00/0.16	 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
0.00/0.16	 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
0.00/0.16	 PROPER(tail(X:S)) -> PROPER(X:S)
0.00/0.16	 PROPER(tail(X:S)) -> TAIL(proper(X:S))
0.00/0.16	 TAIL(mark(X:S)) -> TAIL(X:S)
0.00/0.16	 TAIL(ok(X:S)) -> TAIL(X:S)
0.00/0.16	 TOP(mark(X:S)) -> PROPER(X:S)
0.00/0.16	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.16	 TOP(ok(X:S)) -> ACTIVE(X:S)
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	
0.00/0.16	Problem 1: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
0.00/0.16	 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
0.00/0.16	 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
0.00/0.16	 ACTIVE(tail(X:S)) -> TAIL(active(X:S))
0.00/0.16	 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
0.00/0.16	 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
0.00/0.16	 PROPER(tail(X:S)) -> PROPER(X:S)
0.00/0.16	 PROPER(tail(X:S)) -> TAIL(proper(X:S))
0.00/0.16	 TAIL(mark(X:S)) -> TAIL(X:S)
0.00/0.16	 TAIL(ok(X:S)) -> TAIL(X:S)
0.00/0.16	 TOP(mark(X:S)) -> PROPER(X:S)
0.00/0.16	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.16	 TOP(ok(X:S)) -> ACTIVE(X:S)
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	->->Cycle:
0.00/0.16	->->-> Pairs:
0.00/0.16	 TAIL(mark(X:S)) -> TAIL(X:S)
0.00/0.16	 TAIL(ok(X:S)) -> TAIL(X:S)
0.00/0.16	->->-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->->Cycle:
0.00/0.16	->->-> Pairs:
0.00/0.16	 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
0.00/0.16	 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
0.00/0.16	->->-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->->Cycle:
0.00/0.16	->->-> Pairs:
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
0.00/0.16	 PROPER(tail(X:S)) -> PROPER(X:S)
0.00/0.16	->->-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->->Cycle:
0.00/0.16	->->-> Pairs:
0.00/0.16	 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
0.00/0.16	 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
0.00/0.16	->->-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->->Cycle:
0.00/0.16	->->-> Pairs:
0.00/0.16	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	->->-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	
0.00/0.16	
0.00/0.16	The problem is decomposed in 5 subproblems.
0.00/0.16	
0.00/0.16	Problem 1.1: 
0.00/0.16	
0.00/0.16	Subterm Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 TAIL(mark(X:S)) -> TAIL(X:S)
0.00/0.16	 TAIL(ok(X:S)) -> TAIL(X:S)
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Projection:
0.00/0.16	 pi(TAIL) = 1
0.00/0.16	
0.00/0.16	Problem 1.1: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 Empty
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	 There is no strongly connected component
0.00/0.16	
0.00/0.16	The problem is finite.
0.00/0.16	
0.00/0.16	Problem 1.2: 
0.00/0.16	
0.00/0.16	Subterm Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
0.00/0.16	 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Projection:
0.00/0.16	 pi(CONS) = 1
0.00/0.16	
0.00/0.16	Problem 1.2: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 Empty
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	 There is no strongly connected component
0.00/0.16	
0.00/0.16	The problem is finite.
0.00/0.16	
0.00/0.16	Problem 1.3: 
0.00/0.16	
0.00/0.16	Subterm Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
0.00/0.16	 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
0.00/0.16	 PROPER(tail(X:S)) -> PROPER(X:S)
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Projection:
0.00/0.16	 pi(PROPER) = 1
0.00/0.16	
0.00/0.16	Problem 1.3: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 Empty
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	 There is no strongly connected component
0.00/0.16	
0.00/0.16	The problem is finite.
0.00/0.16	
0.00/0.16	Problem 1.4: 
0.00/0.16	
0.00/0.16	Subterm Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
0.00/0.16	 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Projection:
0.00/0.16	 pi(ACTIVE) = 1
0.00/0.16	
0.00/0.16	Problem 1.4: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 Empty
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	 There is no strongly connected component
0.00/0.16	
0.00/0.16	The problem is finite.
0.00/0.16	
0.00/0.16	Problem 1.5: 
0.00/0.16	
0.00/0.16	Reduction Pairs Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 TOP(mark(X:S)) -> TOP(proper(X:S))
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	-> Usable rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	->Interpretation type:
0.00/0.16	 Linear
0.00/0.16	->Coefficients:
0.00/0.16	 All rationals
0.00/0.16	->Dimension:
0.00/0.16	 1
0.00/0.16	->Bound:
0.00/0.16	 2
0.00/0.16	->Interpretation:
0.00/0.16	 
0.00/0.16	[active](X) = X
0.00/0.16	[cons](X1,X2) = X1 + 1/2.X2
0.00/0.16	[proper](X) = X
0.00/0.16	[tail](X) = 2.X + 1/2
0.00/0.16	[top](X) = 0
0.00/0.16	[0] = 1/2
0.00/0.16	[fSNonEmpty] = 0
0.00/0.16	[mark](X) = X + 1/2
0.00/0.16	[ok](X) = X
0.00/0.16	[zeros] = 2
0.00/0.16	[ACTIVE](X) = 0
0.00/0.16	[CONS](X1,X2) = 0
0.00/0.16	[PROPER](X) = 0
0.00/0.16	[TAIL](X) = 0
0.00/0.16	[TOP](X) = 1/2.X
0.00/0.16	
0.00/0.16	Problem 1.5: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	->->Cycle:
0.00/0.16	->->-> Pairs:
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	->->-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	
0.00/0.16	Problem 1.5: 
0.00/0.16	
0.00/0.16	Reduction Pairs Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 TOP(ok(X:S)) -> TOP(active(X:S))
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	-> Usable rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	->Interpretation type:
0.00/0.16	 Linear
0.00/0.16	->Coefficients:
0.00/0.16	 Natural Numbers
0.00/0.16	->Dimension:
0.00/0.16	 1
0.00/0.16	->Bound:
0.00/0.16	 2
0.00/0.16	->Interpretation:
0.00/0.16	 
0.00/0.16	[active](X) = 2.X
0.00/0.16	[cons](X1,X2) = X1
0.00/0.16	[proper](X) = 0
0.00/0.16	[tail](X) = 2.X + 1
0.00/0.16	[top](X) = 0
0.00/0.16	[0] = 2
0.00/0.16	[fSNonEmpty] = 0
0.00/0.16	[mark](X) = 0
0.00/0.16	[ok](X) = 2.X + 2
0.00/0.16	[zeros] = 0
0.00/0.16	[ACTIVE](X) = 0
0.00/0.16	[CONS](X1,X2) = 0
0.00/0.16	[PROPER](X) = 0
0.00/0.16	[TAIL](X) = 0
0.00/0.16	[TOP](X) = X
0.00/0.16	
0.00/0.16	Problem 1.5: 
0.00/0.16	
0.00/0.16	SCC Processor:
0.00/0.16	-> Pairs:
0.00/0.16	 Empty
0.00/0.16	-> Rules:
0.00/0.16	 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
0.00/0.16	 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
0.00/0.16	 active(tail(X:S)) -> tail(active(X:S))
0.00/0.16	 active(zeros) -> mark(cons(0,zeros))
0.00/0.16	 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
0.00/0.16	 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
0.00/0.16	 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
0.00/0.16	 proper(tail(X:S)) -> tail(proper(X:S))
0.00/0.16	 proper(0) -> ok(0)
0.00/0.16	 proper(zeros) -> ok(zeros)
0.00/0.16	 tail(mark(X:S)) -> mark(tail(X:S))
0.00/0.16	 tail(ok(X:S)) -> ok(tail(X:S))
0.00/0.16	 top(mark(X:S)) -> top(proper(X:S))
0.00/0.16	 top(ok(X:S)) -> top(active(X:S))
0.00/0.16	->Strongly Connected Components:
0.00/0.16	 There is no strongly connected component
0.00/0.16	
0.00/0.16	The problem is finite.
0.00/0.16	EOF