3.63/1.68	WORST_CASE(NON_POLY, ?)
3.63/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.63/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.63/1.69	
3.63/1.69	
3.63/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.63/1.69	
3.63/1.69	(0) CpxTRS
3.63/1.69	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.63/1.69	(2) TRS for Loop Detection
3.63/1.69	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.63/1.69	(4) BEST
3.63/1.69	    (5) proven lower bound
3.63/1.69	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.63/1.69	        (7) BOUNDS(n^1, INF)
3.63/1.69	    (8) TRS for Loop Detection
3.63/1.69	        (9) DecreasingLoopProof [FINISHED, 28 ms]
3.63/1.69	        (10) BOUNDS(EXP, INF)
3.63/1.69	
3.63/1.69	
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(0)
3.63/1.69	Obligation:
3.63/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.63/1.69	
3.63/1.69	
3.63/1.69	The TRS R consists of the following rules:
3.63/1.69	
3.63/1.69	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
3.63/1.69	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
3.63/1.69	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
3.63/1.69	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
3.63/1.69	   nats(N) -> cons(N, n__nats(n__s(N)))
3.63/1.69	   zprimes -> sieve(nats(s(s(0))))
3.63/1.69	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
3.63/1.69	   sieve(X) -> n__sieve(X)
3.63/1.69	   nats(X) -> n__nats(X)
3.63/1.69	   s(X) -> n__s(X)
3.63/1.69	   activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3))
3.63/1.69	   activate(n__sieve(X)) -> sieve(activate(X))
3.63/1.69	   activate(n__nats(X)) -> nats(activate(X))
3.63/1.69	   activate(n__s(X)) -> s(activate(X))
3.63/1.69	   activate(X) -> X
3.63/1.69	
3.63/1.69	S is empty.
3.63/1.69	Rewrite Strategy: FULL
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.63/1.69	Transformed a relative TRS into a decreasing-loop problem.
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(2)
3.63/1.69	Obligation:
3.63/1.69	Analyzing the following TRS for decreasing loops:
3.63/1.69	
3.63/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.63/1.69	
3.63/1.69	
3.63/1.69	The TRS R consists of the following rules:
3.63/1.69	
3.63/1.69	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
3.63/1.69	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
3.63/1.69	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
3.63/1.69	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
3.63/1.69	   nats(N) -> cons(N, n__nats(n__s(N)))
3.63/1.69	   zprimes -> sieve(nats(s(s(0))))
3.63/1.69	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
3.63/1.69	   sieve(X) -> n__sieve(X)
3.63/1.69	   nats(X) -> n__nats(X)
3.63/1.69	   s(X) -> n__s(X)
3.63/1.69	   activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3))
3.63/1.69	   activate(n__sieve(X)) -> sieve(activate(X))
3.63/1.69	   activate(n__nats(X)) -> nats(activate(X))
3.63/1.69	   activate(n__s(X)) -> s(activate(X))
3.63/1.69	   activate(X) -> X
3.63/1.69	
3.63/1.69	S is empty.
3.63/1.69	Rewrite Strategy: FULL
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.63/1.69	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.63/1.69	
3.63/1.69	The rewrite sequence
3.63/1.69	
3.63/1.69	activate(n__s(X)) ->^+ s(activate(X))
3.63/1.69	
3.63/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.63/1.69	
3.63/1.69	The pumping substitution is [X / n__s(X)].
3.63/1.69	
3.63/1.69	The result substitution is [ ].
3.63/1.69	
3.63/1.69	
3.63/1.69	
3.63/1.69	
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(4)
3.63/1.69	Complex Obligation (BEST)
3.63/1.69	
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(5)
3.63/1.69	Obligation:
3.63/1.69	Proved the lower bound n^1 for the following obligation:
3.63/1.69	
3.63/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.63/1.69	
3.63/1.69	
3.63/1.69	The TRS R consists of the following rules:
3.63/1.69	
3.63/1.69	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
3.63/1.69	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
3.63/1.69	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
3.63/1.69	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
3.63/1.69	   nats(N) -> cons(N, n__nats(n__s(N)))
3.63/1.69	   zprimes -> sieve(nats(s(s(0))))
3.63/1.69	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
3.63/1.69	   sieve(X) -> n__sieve(X)
3.63/1.69	   nats(X) -> n__nats(X)
3.63/1.69	   s(X) -> n__s(X)
3.63/1.69	   activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3))
3.63/1.69	   activate(n__sieve(X)) -> sieve(activate(X))
3.63/1.69	   activate(n__nats(X)) -> nats(activate(X))
3.63/1.69	   activate(n__s(X)) -> s(activate(X))
3.63/1.69	   activate(X) -> X
3.63/1.69	
3.63/1.69	S is empty.
3.63/1.69	Rewrite Strategy: FULL
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(6) LowerBoundPropagationProof (FINISHED)
3.63/1.69	Propagated lower bound.
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(7)
3.63/1.69	BOUNDS(n^1, INF)
3.63/1.69	
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(8)
3.63/1.69	Obligation:
3.63/1.69	Analyzing the following TRS for decreasing loops:
3.63/1.69	
3.63/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.63/1.69	
3.63/1.69	
3.63/1.69	The TRS R consists of the following rules:
3.63/1.69	
3.63/1.69	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
3.63/1.69	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
3.63/1.69	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
3.63/1.69	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(n__filter(activate(Y), N, N)))
3.63/1.69	   nats(N) -> cons(N, n__nats(n__s(N)))
3.63/1.69	   zprimes -> sieve(nats(s(s(0))))
3.63/1.69	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
3.63/1.69	   sieve(X) -> n__sieve(X)
3.63/1.69	   nats(X) -> n__nats(X)
3.63/1.69	   s(X) -> n__s(X)
3.63/1.69	   activate(n__filter(X1, X2, X3)) -> filter(activate(X1), activate(X2), activate(X3))
3.63/1.69	   activate(n__sieve(X)) -> sieve(activate(X))
3.63/1.69	   activate(n__nats(X)) -> nats(activate(X))
3.63/1.69	   activate(n__s(X)) -> s(activate(X))
3.63/1.69	   activate(X) -> X
3.63/1.69	
3.63/1.69	S is empty.
3.63/1.69	Rewrite Strategy: FULL
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(9) DecreasingLoopProof (FINISHED)
3.63/1.69	The following loop(s) give(s) rise to the lower bound EXP:
3.63/1.69	
3.63/1.69	The rewrite sequence
3.63/1.69	
3.63/1.69	activate(n__nats(X)) ->^+ cons(activate(X), n__nats(n__s(activate(X))))
3.63/1.69	
3.63/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.63/1.69	
3.63/1.69	The pumping substitution is [X / n__nats(X)].
3.63/1.69	
3.63/1.69	The result substitution is [ ].
3.63/1.69	
3.63/1.69	
3.63/1.69	
3.63/1.69	The rewrite sequence
3.63/1.69	
3.63/1.69	activate(n__nats(X)) ->^+ cons(activate(X), n__nats(n__s(activate(X))))
3.63/1.69	
3.63/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.63/1.69	
3.63/1.69	The pumping substitution is [X / n__nats(X)].
3.63/1.69	
3.63/1.69	The result substitution is [ ].
3.63/1.69	
3.63/1.69	
3.63/1.69	
3.63/1.69	
3.63/1.69	----------------------------------------
3.63/1.69	
3.63/1.69	(10)
3.63/1.69	BOUNDS(EXP, INF)
3.73/1.72	EOF