2.99/1.78	WORST_CASE(NON_POLY, ?)
2.99/1.79	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.99/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.99/1.79	
2.99/1.79	
2.99/1.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.79	
2.99/1.79	(0) CpxTRS
2.99/1.79	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.99/1.79	(2) TRS for Loop Detection
2.99/1.79	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.99/1.79	(4) BEST
2.99/1.79	    (5) proven lower bound
2.99/1.79	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
2.99/1.79	        (7) BOUNDS(n^1, INF)
2.99/1.79	    (8) TRS for Loop Detection
2.99/1.79	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
2.99/1.79	        (10) BOUNDS(INF, INF)
2.99/1.79	
2.99/1.79	
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(0)
2.99/1.79	Obligation:
2.99/1.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.79	
2.99/1.79	
2.99/1.79	The TRS R consists of the following rules:
2.99/1.79	
2.99/1.79	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
2.99/1.79	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
2.99/1.79	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
2.99/1.79	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
2.99/1.79	   nats(N) -> cons(N, nats(s(N)))
2.99/1.79	   zprimes -> sieve(nats(s(s(0))))
2.99/1.79	
2.99/1.79	S is empty.
2.99/1.79	Rewrite Strategy: INNERMOST
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.99/1.79	Transformed a relative TRS into a decreasing-loop problem.
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(2)
2.99/1.79	Obligation:
2.99/1.79	Analyzing the following TRS for decreasing loops:
2.99/1.79	
2.99/1.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.79	
2.99/1.79	
2.99/1.79	The TRS R consists of the following rules:
2.99/1.79	
2.99/1.79	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
2.99/1.79	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
2.99/1.79	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
2.99/1.79	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
2.99/1.79	   nats(N) -> cons(N, nats(s(N)))
2.99/1.79	   zprimes -> sieve(nats(s(s(0))))
2.99/1.79	
2.99/1.79	S is empty.
2.99/1.79	Rewrite Strategy: INNERMOST
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(3) DecreasingLoopProof (LOWER BOUND(ID))
2.99/1.79	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.99/1.79	
2.99/1.79	The rewrite sequence
2.99/1.79	
2.99/1.79	filter(cons(X, Y), s(N), M) ->^+ cons(X, filter(Y, N, M))
2.99/1.79	
2.99/1.79	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
2.99/1.79	
2.99/1.79	The pumping substitution is [Y / cons(X, Y), N / s(N)].
2.99/1.79	
2.99/1.79	The result substitution is [ ].
2.99/1.79	
2.99/1.79	
2.99/1.79	
2.99/1.79	
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(4)
2.99/1.79	Complex Obligation (BEST)
2.99/1.79	
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(5)
2.99/1.79	Obligation:
2.99/1.79	Proved the lower bound n^1 for the following obligation:
2.99/1.79	
2.99/1.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.79	
2.99/1.79	
2.99/1.79	The TRS R consists of the following rules:
2.99/1.79	
2.99/1.79	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
2.99/1.79	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
2.99/1.79	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
2.99/1.79	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
2.99/1.79	   nats(N) -> cons(N, nats(s(N)))
2.99/1.79	   zprimes -> sieve(nats(s(s(0))))
2.99/1.79	
2.99/1.79	S is empty.
2.99/1.79	Rewrite Strategy: INNERMOST
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(6) LowerBoundPropagationProof (FINISHED)
2.99/1.79	Propagated lower bound.
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(7)
2.99/1.79	BOUNDS(n^1, INF)
2.99/1.79	
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(8)
2.99/1.79	Obligation:
2.99/1.79	Analyzing the following TRS for decreasing loops:
2.99/1.79	
2.99/1.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.99/1.79	
2.99/1.79	
2.99/1.79	The TRS R consists of the following rules:
2.99/1.79	
2.99/1.79	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
2.99/1.79	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
2.99/1.79	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
2.99/1.79	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
2.99/1.79	   nats(N) -> cons(N, nats(s(N)))
2.99/1.79	   zprimes -> sieve(nats(s(s(0))))
2.99/1.79	
2.99/1.79	S is empty.
2.99/1.79	Rewrite Strategy: INNERMOST
2.99/1.79	----------------------------------------
2.99/1.79	
2.99/1.79	(9) InfiniteLowerBoundProof (FINISHED)
2.99/1.79	The following loop proves infinite runtime complexity:
2.99/1.79	
2.99/1.79	The rewrite sequence
2.99/1.79	
2.99/1.79	nats(N) ->^+ cons(N, nats(s(N)))
2.99/1.79	
2.99/1.79	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
2.99/1.79	
2.99/1.79	The pumping substitution is [ ].
2.99/1.79	
2.99/1.79	The result substitution is [N / s(N)].
2.99/1.79	
2.99/1.80	
2.99/1.80	
2.99/1.80	
2.99/1.80	----------------------------------------
2.99/1.80	
2.99/1.80	(10)
2.99/1.80	BOUNDS(INF, INF)
3.30/1.83	EOF