3.40/1.71	WORST_CASE(NON_POLY, ?)
3.40/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.40/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.40/1.72	
3.40/1.72	
3.40/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.40/1.72	
3.40/1.72	(0) CpxTRS
3.40/1.72	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.40/1.72	(2) TRS for Loop Detection
3.40/1.72	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.40/1.72	(4) BEST
3.40/1.72	    (5) proven lower bound
3.40/1.72	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.40/1.72	        (7) BOUNDS(n^1, INF)
3.40/1.72	    (8) TRS for Loop Detection
3.40/1.72	        (9) DecreasingLoopProof [FINISHED, 23 ms]
3.40/1.72	        (10) BOUNDS(EXP, INF)
3.40/1.72	
3.40/1.72	
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(0)
3.40/1.72	Obligation:
3.40/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.40/1.72	
3.40/1.72	
3.40/1.72	The TRS R consists of the following rules:
3.40/1.72	
3.40/1.72	   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.40/1.72	   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
3.40/1.72	   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.40/1.72	   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
3.40/1.72	   a__nats(N) -> cons(mark(N), nats(s(N)))
3.40/1.72	   a__zprimes -> a__sieve(a__nats(s(s(0))))
3.40/1.72	   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
3.40/1.72	   mark(sieve(X)) -> a__sieve(mark(X))
3.40/1.72	   mark(nats(X)) -> a__nats(mark(X))
3.40/1.72	   mark(zprimes) -> a__zprimes
3.40/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.40/1.72	   mark(0) -> 0
3.40/1.72	   mark(s(X)) -> s(mark(X))
3.40/1.72	   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
3.40/1.72	   a__sieve(X) -> sieve(X)
3.40/1.72	   a__nats(X) -> nats(X)
3.40/1.72	   a__zprimes -> zprimes
3.40/1.72	
3.40/1.72	S is empty.
3.40/1.72	Rewrite Strategy: FULL
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.40/1.72	Transformed a relative TRS into a decreasing-loop problem.
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(2)
3.40/1.72	Obligation:
3.40/1.72	Analyzing the following TRS for decreasing loops:
3.40/1.72	
3.40/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.40/1.72	
3.40/1.72	
3.40/1.72	The TRS R consists of the following rules:
3.40/1.72	
3.40/1.72	   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.40/1.72	   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
3.40/1.72	   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.40/1.72	   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
3.40/1.72	   a__nats(N) -> cons(mark(N), nats(s(N)))
3.40/1.72	   a__zprimes -> a__sieve(a__nats(s(s(0))))
3.40/1.72	   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
3.40/1.72	   mark(sieve(X)) -> a__sieve(mark(X))
3.40/1.72	   mark(nats(X)) -> a__nats(mark(X))
3.40/1.72	   mark(zprimes) -> a__zprimes
3.40/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.40/1.72	   mark(0) -> 0
3.40/1.72	   mark(s(X)) -> s(mark(X))
3.40/1.72	   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
3.40/1.72	   a__sieve(X) -> sieve(X)
3.40/1.72	   a__nats(X) -> nats(X)
3.40/1.72	   a__zprimes -> zprimes
3.40/1.72	
3.40/1.72	S is empty.
3.40/1.72	Rewrite Strategy: FULL
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.40/1.72	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.40/1.72	
3.40/1.72	The rewrite sequence
3.40/1.72	
3.40/1.72	mark(nats(X)) ->^+ a__nats(mark(X))
3.40/1.72	
3.40/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.40/1.72	
3.40/1.72	The pumping substitution is [X / nats(X)].
3.40/1.72	
3.40/1.72	The result substitution is [ ].
3.40/1.72	
3.40/1.72	
3.40/1.72	
3.40/1.72	
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(4)
3.40/1.72	Complex Obligation (BEST)
3.40/1.72	
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(5)
3.40/1.72	Obligation:
3.40/1.72	Proved the lower bound n^1 for the following obligation:
3.40/1.72	
3.40/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.40/1.72	
3.40/1.72	
3.40/1.72	The TRS R consists of the following rules:
3.40/1.72	
3.40/1.72	   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.40/1.72	   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
3.40/1.72	   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.40/1.72	   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
3.40/1.72	   a__nats(N) -> cons(mark(N), nats(s(N)))
3.40/1.72	   a__zprimes -> a__sieve(a__nats(s(s(0))))
3.40/1.72	   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
3.40/1.72	   mark(sieve(X)) -> a__sieve(mark(X))
3.40/1.72	   mark(nats(X)) -> a__nats(mark(X))
3.40/1.72	   mark(zprimes) -> a__zprimes
3.40/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.40/1.72	   mark(0) -> 0
3.40/1.72	   mark(s(X)) -> s(mark(X))
3.40/1.72	   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
3.40/1.72	   a__sieve(X) -> sieve(X)
3.40/1.72	   a__nats(X) -> nats(X)
3.40/1.72	   a__zprimes -> zprimes
3.40/1.72	
3.40/1.72	S is empty.
3.40/1.72	Rewrite Strategy: FULL
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(6) LowerBoundPropagationProof (FINISHED)
3.40/1.72	Propagated lower bound.
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(7)
3.40/1.72	BOUNDS(n^1, INF)
3.40/1.72	
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(8)
3.40/1.72	Obligation:
3.40/1.72	Analyzing the following TRS for decreasing loops:
3.40/1.72	
3.40/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.40/1.72	
3.40/1.72	
3.40/1.72	The TRS R consists of the following rules:
3.40/1.72	
3.40/1.72	   a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.40/1.72	   a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
3.40/1.72	   a__sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.40/1.72	   a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
3.40/1.72	   a__nats(N) -> cons(mark(N), nats(s(N)))
3.40/1.72	   a__zprimes -> a__sieve(a__nats(s(s(0))))
3.40/1.72	   mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3))
3.40/1.72	   mark(sieve(X)) -> a__sieve(mark(X))
3.40/1.72	   mark(nats(X)) -> a__nats(mark(X))
3.40/1.72	   mark(zprimes) -> a__zprimes
3.40/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.40/1.72	   mark(0) -> 0
3.40/1.72	   mark(s(X)) -> s(mark(X))
3.40/1.72	   a__filter(X1, X2, X3) -> filter(X1, X2, X3)
3.40/1.72	   a__sieve(X) -> sieve(X)
3.40/1.72	   a__nats(X) -> nats(X)
3.40/1.72	   a__zprimes -> zprimes
3.40/1.72	
3.40/1.72	S is empty.
3.40/1.72	Rewrite Strategy: FULL
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(9) DecreasingLoopProof (FINISHED)
3.40/1.72	The following loop(s) give(s) rise to the lower bound EXP:
3.40/1.72	
3.40/1.72	The rewrite sequence
3.40/1.72	
3.40/1.72	mark(nats(X)) ->^+ cons(mark(mark(X)), nats(s(mark(X))))
3.40/1.72	
3.40/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.40/1.72	
3.40/1.72	The pumping substitution is [X / nats(X)].
3.40/1.72	
3.40/1.72	The result substitution is [ ].
3.40/1.72	
3.40/1.72	
3.40/1.72	
3.40/1.72	The rewrite sequence
3.40/1.72	
3.40/1.72	mark(nats(X)) ->^+ cons(mark(mark(X)), nats(s(mark(X))))
3.40/1.72	
3.40/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.40/1.72	
3.40/1.72	The pumping substitution is [X / nats(X)].
3.40/1.72	
3.40/1.72	The result substitution is [ ].
3.40/1.72	
3.40/1.72	
3.40/1.72	
3.40/1.72	
3.40/1.72	----------------------------------------
3.40/1.72	
3.40/1.72	(10)
3.40/1.72	BOUNDS(EXP, INF)
3.73/1.75	EOF