0.00/0.02	YES
0.00/0.02	
0.00/0.02	Problem 1: 
0.00/0.02	
0.00/0.02	(VAR v_NonEmpty:S L:S X:S X1:S X2:S Y:S)
0.00/0.02	(RULES
0.00/0.02	a__eq(0,0) -> ttrue
0.00/0.02	a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	a__eq(X:S,Y:S) -> ffalse
0.00/0.02	a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	a__inf(X:S) -> inf(X:S)
0.00/0.02	a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	a__length(nil) -> 0
0.00/0.02	a__length(X:S) -> length(X:S)
0.00/0.02	a__take(0,X:S) -> nil
0.00/0.02	a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	mark(0) -> 0
0.00/0.02	mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	mark(ffalse) -> ffalse
0.00/0.02	mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	mark(nil) -> nil
0.00/0.02	mark(s(X:S)) -> s(X:S)
0.00/0.02	mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	mark(ttrue) -> ttrue
0.00/0.02	) 
0.00/0.02	(STRATEGY INNERMOST)
0.00/0.02	
0.00/0.02	Problem 1: 
0.00/0.02	
0.00/0.02	Dependency Pairs Processor:
0.00/0.02	-> Pairs:
0.00/0.02	 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
0.00/0.02	 MARK(eq(X1:S,X2:S)) -> A__EQ(X1:S,X2:S)
0.00/0.02	 MARK(inf(X:S)) -> A__INF(mark(X:S))
0.00/0.02	 MARK(inf(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(length(X:S)) -> A__LENGTH(mark(X:S))
0.00/0.02	 MARK(length(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S))
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
0.00/0.02	-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	
0.00/0.02	Problem 1: 
0.00/0.02	
0.00/0.02	SCC Processor:
0.00/0.02	-> Pairs:
0.00/0.02	 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
0.00/0.02	 MARK(eq(X1:S,X2:S)) -> A__EQ(X1:S,X2:S)
0.00/0.02	 MARK(inf(X:S)) -> A__INF(mark(X:S))
0.00/0.02	 MARK(inf(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(length(X:S)) -> A__LENGTH(mark(X:S))
0.00/0.02	 MARK(length(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S))
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
0.00/0.02	-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	->Strongly Connected Components:
0.00/0.02	->->Cycle:
0.00/0.02	->->-> Pairs:
0.00/0.02	 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
0.00/0.02	->->-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	->->Cycle:
0.00/0.02	->->-> Pairs:
0.00/0.02	 MARK(inf(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(length(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
0.00/0.02	->->-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	
0.00/0.02	
0.00/0.02	The problem is decomposed in 2 subproblems.
0.00/0.02	
0.00/0.02	Problem 1.1: 
0.00/0.02	
0.00/0.02	Subterm Processor:
0.00/0.02	-> Pairs:
0.00/0.02	 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
0.00/0.02	-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	->Projection:
0.00/0.02	 pi(A__EQ) = 1
0.00/0.02	
0.00/0.02	Problem 1.1: 
0.00/0.02	
0.00/0.02	SCC Processor:
0.00/0.02	-> Pairs:
0.00/0.02	 Empty
0.00/0.02	-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	->Strongly Connected Components:
0.00/0.02	 There is no strongly connected component
0.00/0.02	
0.00/0.02	The problem is finite.
0.00/0.02	
0.00/0.02	Problem 1.2: 
0.00/0.02	
0.00/0.02	Subterm Processor:
0.00/0.02	-> Pairs:
0.00/0.02	 MARK(inf(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(length(X:S)) -> MARK(X:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
0.00/0.02	 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
0.00/0.02	-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	->Projection:
0.00/0.02	 pi(MARK) = 1
0.00/0.02	
0.00/0.02	Problem 1.2: 
0.00/0.02	
0.00/0.02	SCC Processor:
0.00/0.02	-> Pairs:
0.00/0.02	 Empty
0.00/0.02	-> Rules:
0.00/0.02	 a__eq(0,0) -> ttrue
0.00/0.02	 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
0.00/0.02	 a__eq(X:S,Y:S) -> ffalse
0.00/0.02	 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
0.00/0.02	 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
0.00/0.02	 a__inf(X:S) -> inf(X:S)
0.00/0.02	 a__length(cons(X:S,L:S)) -> s(length(L:S))
0.00/0.02	 a__length(nil) -> 0
0.00/0.02	 a__length(X:S) -> length(X:S)
0.00/0.02	 a__take(0,X:S) -> nil
0.00/0.02	 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
0.00/0.02	 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
0.00/0.02	 mark(0) -> 0
0.00/0.02	 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
0.00/0.02	 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
0.00/0.02	 mark(ffalse) -> ffalse
0.00/0.02	 mark(inf(X:S)) -> a__inf(mark(X:S))
0.00/0.02	 mark(length(X:S)) -> a__length(mark(X:S))
0.00/0.02	 mark(nil) -> nil
0.00/0.02	 mark(s(X:S)) -> s(X:S)
0.00/0.02	 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
0.00/0.02	 mark(ttrue) -> ttrue
0.00/0.02	->Strongly Connected Components:
0.00/0.02	 There is no strongly connected component
0.00/0.02	
0.00/0.02	The problem is finite.
0.00/0.02	EOF