2.97/1.56	WORST_CASE(NON_POLY, ?)
3.34/1.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.34/1.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.34/1.56	
3.34/1.56	
3.34/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.56	
3.34/1.56	(0) CpxTRS
3.34/1.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.34/1.56	(2) TRS for Loop Detection
3.34/1.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.34/1.56	(4) BEST
3.34/1.56	    (5) proven lower bound
3.34/1.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.34/1.56	        (7) BOUNDS(n^1, INF)
3.34/1.56	    (8) TRS for Loop Detection
3.34/1.56	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.34/1.56	        (10) BOUNDS(EXP, INF)
3.34/1.56	
3.34/1.56	
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(0)
3.34/1.56	Obligation:
3.34/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.56	
3.34/1.56	
3.34/1.56	The TRS R consists of the following rules:
3.34/1.56	
3.34/1.56	   a__first(0, X) -> nil
3.34/1.56	   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
3.34/1.56	   a__from(X) -> cons(mark(X), from(s(X)))
3.34/1.56	   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
3.34/1.56	   mark(from(X)) -> a__from(mark(X))
3.34/1.56	   mark(0) -> 0
3.34/1.56	   mark(nil) -> nil
3.34/1.56	   mark(s(X)) -> s(mark(X))
3.34/1.56	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.34/1.56	   a__first(X1, X2) -> first(X1, X2)
3.34/1.56	   a__from(X) -> from(X)
3.34/1.56	
3.34/1.56	S is empty.
3.34/1.56	Rewrite Strategy: FULL
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.34/1.56	Transformed a relative TRS into a decreasing-loop problem.
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(2)
3.34/1.56	Obligation:
3.34/1.56	Analyzing the following TRS for decreasing loops:
3.34/1.56	
3.34/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.56	
3.34/1.56	
3.34/1.56	The TRS R consists of the following rules:
3.34/1.56	
3.34/1.56	   a__first(0, X) -> nil
3.34/1.56	   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
3.34/1.56	   a__from(X) -> cons(mark(X), from(s(X)))
3.34/1.56	   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
3.34/1.56	   mark(from(X)) -> a__from(mark(X))
3.34/1.56	   mark(0) -> 0
3.34/1.56	   mark(nil) -> nil
3.34/1.56	   mark(s(X)) -> s(mark(X))
3.34/1.56	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.34/1.56	   a__first(X1, X2) -> first(X1, X2)
3.34/1.56	   a__from(X) -> from(X)
3.34/1.56	
3.34/1.56	S is empty.
3.34/1.56	Rewrite Strategy: FULL
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.34/1.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.34/1.56	
3.34/1.56	The rewrite sequence
3.34/1.56	
3.34/1.56	mark(from(X)) ->^+ a__from(mark(X))
3.34/1.56	
3.34/1.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.34/1.56	
3.34/1.56	The pumping substitution is [X / from(X)].
3.34/1.56	
3.34/1.56	The result substitution is [ ].
3.34/1.56	
3.34/1.56	
3.34/1.56	
3.34/1.56	
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(4)
3.34/1.56	Complex Obligation (BEST)
3.34/1.56	
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(5)
3.34/1.56	Obligation:
3.34/1.56	Proved the lower bound n^1 for the following obligation:
3.34/1.56	
3.34/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.56	
3.34/1.56	
3.34/1.56	The TRS R consists of the following rules:
3.34/1.56	
3.34/1.56	   a__first(0, X) -> nil
3.34/1.56	   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
3.34/1.56	   a__from(X) -> cons(mark(X), from(s(X)))
3.34/1.56	   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
3.34/1.56	   mark(from(X)) -> a__from(mark(X))
3.34/1.56	   mark(0) -> 0
3.34/1.56	   mark(nil) -> nil
3.34/1.56	   mark(s(X)) -> s(mark(X))
3.34/1.56	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.34/1.56	   a__first(X1, X2) -> first(X1, X2)
3.34/1.56	   a__from(X) -> from(X)
3.34/1.56	
3.34/1.56	S is empty.
3.34/1.56	Rewrite Strategy: FULL
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(6) LowerBoundPropagationProof (FINISHED)
3.34/1.56	Propagated lower bound.
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(7)
3.34/1.56	BOUNDS(n^1, INF)
3.34/1.56	
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(8)
3.34/1.56	Obligation:
3.34/1.56	Analyzing the following TRS for decreasing loops:
3.34/1.56	
3.34/1.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.56	
3.34/1.56	
3.34/1.56	The TRS R consists of the following rules:
3.34/1.56	
3.34/1.56	   a__first(0, X) -> nil
3.34/1.56	   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
3.34/1.56	   a__from(X) -> cons(mark(X), from(s(X)))
3.34/1.56	   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
3.34/1.56	   mark(from(X)) -> a__from(mark(X))
3.34/1.56	   mark(0) -> 0
3.34/1.56	   mark(nil) -> nil
3.34/1.56	   mark(s(X)) -> s(mark(X))
3.34/1.56	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.34/1.56	   a__first(X1, X2) -> first(X1, X2)
3.34/1.56	   a__from(X) -> from(X)
3.34/1.56	
3.34/1.56	S is empty.
3.34/1.56	Rewrite Strategy: FULL
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(9) DecreasingLoopProof (FINISHED)
3.34/1.56	The following loop(s) give(s) rise to the lower bound EXP:
3.34/1.56	
3.34/1.56	The rewrite sequence
3.34/1.56	
3.34/1.56	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.34/1.56	
3.34/1.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.34/1.56	
3.34/1.56	The pumping substitution is [X / from(X)].
3.34/1.56	
3.34/1.56	The result substitution is [ ].
3.34/1.56	
3.34/1.56	
3.34/1.56	
3.34/1.56	The rewrite sequence
3.34/1.56	
3.34/1.56	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.34/1.56	
3.34/1.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.34/1.56	
3.34/1.56	The pumping substitution is [X / from(X)].
3.34/1.56	
3.34/1.56	The result substitution is [ ].
3.34/1.56	
3.34/1.56	
3.34/1.56	
3.34/1.56	
3.34/1.56	----------------------------------------
3.34/1.56	
3.34/1.56	(10)
3.34/1.56	BOUNDS(EXP, INF)
3.34/1.59	EOF