3.76/1.83	WORST_CASE(NON_POLY, ?)
3.76/1.84	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.76/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.76/1.84	
3.76/1.84	
3.76/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.76/1.84	
3.76/1.84	(0) CpxTRS
3.76/1.84	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.76/1.84	(2) TRS for Loop Detection
3.76/1.84	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.76/1.84	(4) BEST
3.76/1.84	    (5) proven lower bound
3.76/1.84	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.76/1.84	        (7) BOUNDS(n^1, INF)
3.76/1.84	    (8) TRS for Loop Detection
3.76/1.84	        (9) DecreasingLoopProof [FINISHED, 70 ms]
3.76/1.84	        (10) BOUNDS(EXP, INF)
3.76/1.84	
3.76/1.84	
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(0)
3.76/1.84	Obligation:
3.76/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.76/1.84	
3.76/1.84	
3.76/1.84	The TRS R consists of the following rules:
3.76/1.84	
3.76/1.84	   a__from(X) -> cons(mark(X), from(s(X)))
3.76/1.84	   a__sel(0, cons(X, XS)) -> mark(X)
3.76/1.84	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.76/1.84	   a__minus(X, 0) -> 0
3.76/1.84	   a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y))
3.76/1.84	   a__quot(0, s(Y)) -> 0
3.76/1.84	   a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
3.76/1.84	   a__zWquot(XS, nil) -> nil
3.76/1.84	   a__zWquot(nil, XS) -> nil
3.76/1.84	   a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
3.76/1.84	   mark(from(X)) -> a__from(mark(X))
3.76/1.84	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.76/1.84	   mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2))
3.76/1.84	   mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2))
3.76/1.84	   mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2))
3.76/1.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.76/1.84	   mark(s(X)) -> s(mark(X))
3.76/1.84	   mark(0) -> 0
3.76/1.84	   mark(nil) -> nil
3.76/1.84	   a__from(X) -> from(X)
3.76/1.84	   a__sel(X1, X2) -> sel(X1, X2)
3.76/1.84	   a__minus(X1, X2) -> minus(X1, X2)
3.76/1.84	   a__quot(X1, X2) -> quot(X1, X2)
3.76/1.84	   a__zWquot(X1, X2) -> zWquot(X1, X2)
3.76/1.84	
3.76/1.84	S is empty.
3.76/1.84	Rewrite Strategy: FULL
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.76/1.84	Transformed a relative TRS into a decreasing-loop problem.
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(2)
3.76/1.84	Obligation:
3.76/1.84	Analyzing the following TRS for decreasing loops:
3.76/1.84	
3.76/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.76/1.84	
3.76/1.84	
3.76/1.84	The TRS R consists of the following rules:
3.76/1.84	
3.76/1.84	   a__from(X) -> cons(mark(X), from(s(X)))
3.76/1.84	   a__sel(0, cons(X, XS)) -> mark(X)
3.76/1.84	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.76/1.84	   a__minus(X, 0) -> 0
3.76/1.84	   a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y))
3.76/1.84	   a__quot(0, s(Y)) -> 0
3.76/1.84	   a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
3.76/1.84	   a__zWquot(XS, nil) -> nil
3.76/1.84	   a__zWquot(nil, XS) -> nil
3.76/1.84	   a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
3.76/1.84	   mark(from(X)) -> a__from(mark(X))
3.76/1.84	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.76/1.84	   mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2))
3.76/1.84	   mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2))
3.76/1.84	   mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2))
3.76/1.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.76/1.84	   mark(s(X)) -> s(mark(X))
3.76/1.84	   mark(0) -> 0
3.76/1.84	   mark(nil) -> nil
3.76/1.84	   a__from(X) -> from(X)
3.76/1.84	   a__sel(X1, X2) -> sel(X1, X2)
3.76/1.84	   a__minus(X1, X2) -> minus(X1, X2)
3.76/1.84	   a__quot(X1, X2) -> quot(X1, X2)
3.76/1.84	   a__zWquot(X1, X2) -> zWquot(X1, X2)
3.76/1.84	
3.76/1.84	S is empty.
3.76/1.84	Rewrite Strategy: FULL
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.76/1.84	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.76/1.84	
3.76/1.84	The rewrite sequence
3.76/1.84	
3.76/1.84	mark(from(X)) ->^+ a__from(mark(X))
3.76/1.84	
3.76/1.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.76/1.84	
3.76/1.84	The pumping substitution is [X / from(X)].
3.76/1.84	
3.76/1.84	The result substitution is [ ].
3.76/1.84	
3.76/1.84	
3.76/1.84	
3.76/1.84	
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(4)
3.76/1.84	Complex Obligation (BEST)
3.76/1.84	
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(5)
3.76/1.84	Obligation:
3.76/1.84	Proved the lower bound n^1 for the following obligation:
3.76/1.84	
3.76/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.76/1.84	
3.76/1.84	
3.76/1.84	The TRS R consists of the following rules:
3.76/1.84	
3.76/1.84	   a__from(X) -> cons(mark(X), from(s(X)))
3.76/1.84	   a__sel(0, cons(X, XS)) -> mark(X)
3.76/1.84	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.76/1.84	   a__minus(X, 0) -> 0
3.76/1.84	   a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y))
3.76/1.84	   a__quot(0, s(Y)) -> 0
3.76/1.84	   a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
3.76/1.84	   a__zWquot(XS, nil) -> nil
3.76/1.84	   a__zWquot(nil, XS) -> nil
3.76/1.84	   a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
3.76/1.84	   mark(from(X)) -> a__from(mark(X))
3.76/1.84	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.76/1.84	   mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2))
3.76/1.84	   mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2))
3.76/1.84	   mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2))
3.76/1.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.76/1.84	   mark(s(X)) -> s(mark(X))
3.76/1.84	   mark(0) -> 0
3.76/1.84	   mark(nil) -> nil
3.76/1.84	   a__from(X) -> from(X)
3.76/1.84	   a__sel(X1, X2) -> sel(X1, X2)
3.76/1.84	   a__minus(X1, X2) -> minus(X1, X2)
3.76/1.84	   a__quot(X1, X2) -> quot(X1, X2)
3.76/1.84	   a__zWquot(X1, X2) -> zWquot(X1, X2)
3.76/1.84	
3.76/1.84	S is empty.
3.76/1.84	Rewrite Strategy: FULL
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(6) LowerBoundPropagationProof (FINISHED)
3.76/1.84	Propagated lower bound.
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(7)
3.76/1.84	BOUNDS(n^1, INF)
3.76/1.84	
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(8)
3.76/1.84	Obligation:
3.76/1.84	Analyzing the following TRS for decreasing loops:
3.76/1.84	
3.76/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.76/1.84	
3.76/1.84	
3.76/1.84	The TRS R consists of the following rules:
3.76/1.84	
3.76/1.84	   a__from(X) -> cons(mark(X), from(s(X)))
3.76/1.84	   a__sel(0, cons(X, XS)) -> mark(X)
3.76/1.84	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.76/1.84	   a__minus(X, 0) -> 0
3.76/1.84	   a__minus(s(X), s(Y)) -> a__minus(mark(X), mark(Y))
3.76/1.84	   a__quot(0, s(Y)) -> 0
3.76/1.84	   a__quot(s(X), s(Y)) -> s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
3.76/1.84	   a__zWquot(XS, nil) -> nil
3.76/1.84	   a__zWquot(nil, XS) -> nil
3.76/1.84	   a__zWquot(cons(X, XS), cons(Y, YS)) -> cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
3.76/1.84	   mark(from(X)) -> a__from(mark(X))
3.76/1.84	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.76/1.84	   mark(minus(X1, X2)) -> a__minus(mark(X1), mark(X2))
3.76/1.84	   mark(quot(X1, X2)) -> a__quot(mark(X1), mark(X2))
3.76/1.84	   mark(zWquot(X1, X2)) -> a__zWquot(mark(X1), mark(X2))
3.76/1.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.76/1.84	   mark(s(X)) -> s(mark(X))
3.76/1.84	   mark(0) -> 0
3.76/1.84	   mark(nil) -> nil
3.76/1.84	   a__from(X) -> from(X)
3.76/1.84	   a__sel(X1, X2) -> sel(X1, X2)
3.76/1.84	   a__minus(X1, X2) -> minus(X1, X2)
3.76/1.84	   a__quot(X1, X2) -> quot(X1, X2)
3.76/1.84	   a__zWquot(X1, X2) -> zWquot(X1, X2)
3.76/1.84	
3.76/1.84	S is empty.
3.76/1.84	Rewrite Strategy: FULL
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(9) DecreasingLoopProof (FINISHED)
3.76/1.84	The following loop(s) give(s) rise to the lower bound EXP:
3.76/1.84	
3.76/1.84	The rewrite sequence
3.76/1.84	
3.76/1.84	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.76/1.84	
3.76/1.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.76/1.84	
3.76/1.84	The pumping substitution is [X / from(X)].
3.76/1.84	
3.76/1.84	The result substitution is [ ].
3.76/1.84	
3.76/1.84	
3.76/1.84	
3.76/1.84	The rewrite sequence
3.76/1.84	
3.76/1.84	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.76/1.84	
3.76/1.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.76/1.84	
3.76/1.84	The pumping substitution is [X / from(X)].
3.76/1.84	
3.76/1.84	The result substitution is [ ].
3.76/1.84	
3.76/1.84	
3.76/1.84	
3.76/1.84	
3.76/1.84	----------------------------------------
3.76/1.84	
3.76/1.84	(10)
3.76/1.84	BOUNDS(EXP, INF)
3.93/1.88	EOF