2.95/1.51	WORST_CASE(NON_POLY, ?)
3.09/1.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.09/1.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.09/1.52	
3.09/1.52	
3.09/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.09/1.52	
3.09/1.52	(0) CpxTRS
3.09/1.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.09/1.52	(2) TRS for Loop Detection
3.09/1.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.09/1.52	(4) BEST
3.09/1.52	    (5) proven lower bound
3.09/1.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.09/1.52	        (7) BOUNDS(n^1, INF)
3.09/1.52	    (8) TRS for Loop Detection
3.09/1.52	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.09/1.52	        (10) BOUNDS(INF, INF)
3.09/1.52	
3.09/1.52	
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(0)
3.09/1.52	Obligation:
3.09/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.09/1.52	
3.09/1.52	
3.09/1.52	The TRS R consists of the following rules:
3.09/1.52	
3.09/1.52	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.09/1.52	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.09/1.52	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.09/1.52	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.09/1.52	   nats(N) -> cons(N, nats(s(N)))
3.09/1.52	   zprimes -> sieve(nats(s(s(0))))
3.09/1.52	
3.09/1.52	S is empty.
3.09/1.52	Rewrite Strategy: FULL
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.09/1.52	Transformed a relative TRS into a decreasing-loop problem.
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(2)
3.09/1.52	Obligation:
3.09/1.52	Analyzing the following TRS for decreasing loops:
3.09/1.52	
3.09/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.09/1.52	
3.09/1.52	
3.09/1.52	The TRS R consists of the following rules:
3.09/1.52	
3.09/1.52	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.09/1.52	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.09/1.52	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.09/1.52	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.09/1.52	   nats(N) -> cons(N, nats(s(N)))
3.09/1.52	   zprimes -> sieve(nats(s(s(0))))
3.09/1.52	
3.09/1.52	S is empty.
3.09/1.52	Rewrite Strategy: FULL
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.09/1.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.09/1.52	
3.09/1.52	The rewrite sequence
3.09/1.52	
3.09/1.52	filter(cons(X, Y), s(N), M) ->^+ cons(X, filter(Y, N, M))
3.09/1.52	
3.09/1.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.09/1.52	
3.09/1.52	The pumping substitution is [Y / cons(X, Y), N / s(N)].
3.09/1.52	
3.09/1.52	The result substitution is [ ].
3.09/1.52	
3.09/1.52	
3.09/1.52	
3.09/1.52	
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(4)
3.09/1.52	Complex Obligation (BEST)
3.09/1.52	
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(5)
3.09/1.52	Obligation:
3.09/1.52	Proved the lower bound n^1 for the following obligation:
3.09/1.52	
3.09/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.09/1.52	
3.09/1.52	
3.09/1.52	The TRS R consists of the following rules:
3.09/1.52	
3.09/1.52	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.09/1.52	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.09/1.52	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.09/1.52	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.09/1.52	   nats(N) -> cons(N, nats(s(N)))
3.09/1.52	   zprimes -> sieve(nats(s(s(0))))
3.09/1.52	
3.09/1.52	S is empty.
3.09/1.52	Rewrite Strategy: FULL
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(6) LowerBoundPropagationProof (FINISHED)
3.09/1.52	Propagated lower bound.
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(7)
3.09/1.52	BOUNDS(n^1, INF)
3.09/1.52	
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(8)
3.09/1.52	Obligation:
3.09/1.52	Analyzing the following TRS for decreasing loops:
3.09/1.52	
3.09/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.09/1.52	
3.09/1.52	
3.09/1.52	The TRS R consists of the following rules:
3.09/1.52	
3.09/1.52	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.09/1.52	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.09/1.52	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.09/1.52	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.09/1.52	   nats(N) -> cons(N, nats(s(N)))
3.09/1.52	   zprimes -> sieve(nats(s(s(0))))
3.09/1.52	
3.09/1.52	S is empty.
3.09/1.52	Rewrite Strategy: FULL
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(9) InfiniteLowerBoundProof (FINISHED)
3.09/1.52	The following loop proves infinite runtime complexity:
3.09/1.52	
3.09/1.52	The rewrite sequence
3.09/1.52	
3.09/1.52	nats(N) ->^+ cons(N, nats(s(N)))
3.09/1.52	
3.09/1.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.09/1.52	
3.09/1.52	The pumping substitution is [ ].
3.09/1.52	
3.09/1.52	The result substitution is [N / s(N)].
3.09/1.52	
3.09/1.52	
3.09/1.52	
3.09/1.52	
3.09/1.52	----------------------------------------
3.09/1.52	
3.09/1.52	(10)
3.09/1.52	BOUNDS(INF, INF)
3.12/1.55	EOF