6.01/2.34	WORST_CASE(NON_POLY, ?)
6.01/2.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.01/2.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.01/2.35	
6.01/2.35	
6.01/2.35	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.01/2.35	
6.01/2.35	(0) CpxTRS
6.01/2.35	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
6.01/2.35	(2) TRS for Loop Detection
6.01/2.35	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
6.01/2.35	(4) BEST
6.01/2.35	    (5) proven lower bound
6.01/2.35	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
6.01/2.35	        (7) BOUNDS(n^1, INF)
6.01/2.35	    (8) TRS for Loop Detection
6.01/2.35	        (9) DecreasingLoopProof [FINISHED, 472 ms]
6.01/2.35	        (10) BOUNDS(EXP, INF)
6.01/2.35	
6.01/2.35	
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(0)
6.01/2.35	Obligation:
6.01/2.35	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.01/2.35	
6.01/2.35	
6.01/2.35	The TRS R consists of the following rules:
6.01/2.35	
6.01/2.35	   a__pairNs -> cons(0, incr(oddNs))
6.01/2.35	   a__oddNs -> a__incr(a__pairNs)
6.01/2.35	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
6.01/2.35	   a__take(0, XS) -> nil
6.01/2.35	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
6.01/2.35	   a__zip(nil, XS) -> nil
6.01/2.35	   a__zip(X, nil) -> nil
6.01/2.35	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
6.01/2.35	   a__tail(cons(X, XS)) -> mark(XS)
6.01/2.35	   a__repItems(nil) -> nil
6.01/2.35	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
6.01/2.35	   mark(pairNs) -> a__pairNs
6.01/2.35	   mark(incr(X)) -> a__incr(mark(X))
6.01/2.35	   mark(oddNs) -> a__oddNs
6.01/2.35	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
6.01/2.35	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
6.01/2.35	   mark(tail(X)) -> a__tail(mark(X))
6.01/2.35	   mark(repItems(X)) -> a__repItems(mark(X))
6.01/2.35	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
6.01/2.35	   mark(0) -> 0
6.01/2.35	   mark(s(X)) -> s(mark(X))
6.01/2.35	   mark(nil) -> nil
6.01/2.35	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
6.01/2.35	   a__pairNs -> pairNs
6.01/2.35	   a__incr(X) -> incr(X)
6.01/2.35	   a__oddNs -> oddNs
6.01/2.35	   a__take(X1, X2) -> take(X1, X2)
6.01/2.35	   a__zip(X1, X2) -> zip(X1, X2)
6.01/2.35	   a__tail(X) -> tail(X)
6.01/2.35	   a__repItems(X) -> repItems(X)
6.01/2.35	
6.01/2.35	S is empty.
6.01/2.35	Rewrite Strategy: FULL
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
6.01/2.35	Transformed a relative TRS into a decreasing-loop problem.
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(2)
6.01/2.35	Obligation:
6.01/2.35	Analyzing the following TRS for decreasing loops:
6.01/2.35	
6.01/2.35	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.01/2.35	
6.01/2.35	
6.01/2.35	The TRS R consists of the following rules:
6.01/2.35	
6.01/2.35	   a__pairNs -> cons(0, incr(oddNs))
6.01/2.35	   a__oddNs -> a__incr(a__pairNs)
6.01/2.35	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
6.01/2.35	   a__take(0, XS) -> nil
6.01/2.35	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
6.01/2.35	   a__zip(nil, XS) -> nil
6.01/2.35	   a__zip(X, nil) -> nil
6.01/2.35	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
6.01/2.35	   a__tail(cons(X, XS)) -> mark(XS)
6.01/2.35	   a__repItems(nil) -> nil
6.01/2.35	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
6.01/2.35	   mark(pairNs) -> a__pairNs
6.01/2.35	   mark(incr(X)) -> a__incr(mark(X))
6.01/2.35	   mark(oddNs) -> a__oddNs
6.01/2.35	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
6.01/2.35	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
6.01/2.35	   mark(tail(X)) -> a__tail(mark(X))
6.01/2.35	   mark(repItems(X)) -> a__repItems(mark(X))
6.01/2.35	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
6.01/2.35	   mark(0) -> 0
6.01/2.35	   mark(s(X)) -> s(mark(X))
6.01/2.35	   mark(nil) -> nil
6.01/2.35	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
6.01/2.35	   a__pairNs -> pairNs
6.01/2.35	   a__incr(X) -> incr(X)
6.01/2.35	   a__oddNs -> oddNs
6.01/2.35	   a__take(X1, X2) -> take(X1, X2)
6.01/2.35	   a__zip(X1, X2) -> zip(X1, X2)
6.01/2.35	   a__tail(X) -> tail(X)
6.01/2.35	   a__repItems(X) -> repItems(X)
6.01/2.35	
6.01/2.35	S is empty.
6.01/2.35	Rewrite Strategy: FULL
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(3) DecreasingLoopProof (LOWER BOUND(ID))
6.01/2.35	The following loop(s) give(s) rise to the lower bound Omega(n^1):
6.01/2.35	
6.01/2.35	The rewrite sequence
6.01/2.35	
6.01/2.35	mark(repItems(X)) ->^+ a__repItems(mark(X))
6.01/2.35	
6.01/2.35	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
6.01/2.35	
6.01/2.35	The pumping substitution is [X / repItems(X)].
6.01/2.35	
6.01/2.35	The result substitution is [ ].
6.01/2.35	
6.01/2.35	
6.01/2.35	
6.01/2.35	
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(4)
6.01/2.35	Complex Obligation (BEST)
6.01/2.35	
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(5)
6.01/2.35	Obligation:
6.01/2.35	Proved the lower bound n^1 for the following obligation:
6.01/2.35	
6.01/2.35	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.01/2.35	
6.01/2.35	
6.01/2.35	The TRS R consists of the following rules:
6.01/2.35	
6.01/2.35	   a__pairNs -> cons(0, incr(oddNs))
6.01/2.35	   a__oddNs -> a__incr(a__pairNs)
6.01/2.35	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
6.01/2.35	   a__take(0, XS) -> nil
6.01/2.35	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
6.01/2.35	   a__zip(nil, XS) -> nil
6.01/2.35	   a__zip(X, nil) -> nil
6.01/2.35	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
6.01/2.35	   a__tail(cons(X, XS)) -> mark(XS)
6.01/2.35	   a__repItems(nil) -> nil
6.01/2.35	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
6.01/2.35	   mark(pairNs) -> a__pairNs
6.01/2.35	   mark(incr(X)) -> a__incr(mark(X))
6.01/2.35	   mark(oddNs) -> a__oddNs
6.01/2.35	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
6.01/2.35	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
6.01/2.35	   mark(tail(X)) -> a__tail(mark(X))
6.01/2.35	   mark(repItems(X)) -> a__repItems(mark(X))
6.01/2.35	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
6.01/2.35	   mark(0) -> 0
6.01/2.35	   mark(s(X)) -> s(mark(X))
6.01/2.35	   mark(nil) -> nil
6.01/2.35	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
6.01/2.35	   a__pairNs -> pairNs
6.01/2.35	   a__incr(X) -> incr(X)
6.01/2.35	   a__oddNs -> oddNs
6.01/2.35	   a__take(X1, X2) -> take(X1, X2)
6.01/2.35	   a__zip(X1, X2) -> zip(X1, X2)
6.01/2.35	   a__tail(X) -> tail(X)
6.01/2.35	   a__repItems(X) -> repItems(X)
6.01/2.35	
6.01/2.35	S is empty.
6.01/2.35	Rewrite Strategy: FULL
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(6) LowerBoundPropagationProof (FINISHED)
6.01/2.35	Propagated lower bound.
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(7)
6.01/2.35	BOUNDS(n^1, INF)
6.01/2.35	
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(8)
6.01/2.35	Obligation:
6.01/2.35	Analyzing the following TRS for decreasing loops:
6.01/2.35	
6.01/2.35	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.01/2.35	
6.01/2.35	
6.01/2.35	The TRS R consists of the following rules:
6.01/2.35	
6.01/2.35	   a__pairNs -> cons(0, incr(oddNs))
6.01/2.35	   a__oddNs -> a__incr(a__pairNs)
6.01/2.35	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
6.01/2.35	   a__take(0, XS) -> nil
6.01/2.35	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
6.01/2.35	   a__zip(nil, XS) -> nil
6.01/2.35	   a__zip(X, nil) -> nil
6.01/2.35	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
6.01/2.35	   a__tail(cons(X, XS)) -> mark(XS)
6.01/2.35	   a__repItems(nil) -> nil
6.01/2.35	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
6.01/2.35	   mark(pairNs) -> a__pairNs
6.01/2.35	   mark(incr(X)) -> a__incr(mark(X))
6.01/2.35	   mark(oddNs) -> a__oddNs
6.01/2.35	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
6.01/2.35	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
6.01/2.35	   mark(tail(X)) -> a__tail(mark(X))
6.01/2.35	   mark(repItems(X)) -> a__repItems(mark(X))
6.01/2.35	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
6.01/2.35	   mark(0) -> 0
6.01/2.35	   mark(s(X)) -> s(mark(X))
6.01/2.35	   mark(nil) -> nil
6.01/2.35	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
6.01/2.35	   a__pairNs -> pairNs
6.01/2.35	   a__incr(X) -> incr(X)
6.01/2.35	   a__oddNs -> oddNs
6.01/2.35	   a__take(X1, X2) -> take(X1, X2)
6.01/2.35	   a__zip(X1, X2) -> zip(X1, X2)
6.01/2.35	   a__tail(X) -> tail(X)
6.01/2.35	   a__repItems(X) -> repItems(X)
6.01/2.35	
6.01/2.35	S is empty.
6.01/2.35	Rewrite Strategy: FULL
6.01/2.35	----------------------------------------
6.01/2.35	
6.01/2.35	(9) DecreasingLoopProof (FINISHED)
6.01/2.35	The following loop(s) give(s) rise to the lower bound EXP:
6.01/2.35	
6.01/2.35	The rewrite sequence
6.01/2.35	
6.01/2.35	mark(repItems(cons(X11_0, X22_0))) ->^+ cons(mark(mark(X11_0)), cons(mark(X11_0), repItems(X22_0)))
6.01/2.36	
6.01/2.36	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
6.01/2.36	
6.01/2.36	The pumping substitution is [X11_0 / repItems(cons(X11_0, X22_0))].
6.01/2.36	
6.01/2.36	The result substitution is [ ].
6.01/2.36	
6.01/2.36	
6.01/2.36	
6.01/2.36	The rewrite sequence
6.01/2.36	
6.01/2.36	mark(repItems(cons(X11_0, X22_0))) ->^+ cons(mark(mark(X11_0)), cons(mark(X11_0), repItems(X22_0)))
6.01/2.36	
6.01/2.36	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
6.01/2.36	
6.01/2.36	The pumping substitution is [X11_0 / repItems(cons(X11_0, X22_0))].
6.01/2.36	
6.01/2.36	The result substitution is [ ].
6.01/2.36	
6.01/2.36	
6.01/2.36	
6.01/2.36	
6.01/2.36	----------------------------------------
6.01/2.36	
6.01/2.36	(10)
6.01/2.36	BOUNDS(EXP, INF)
6.09/2.40	EOF