323.38/291.53	WORST_CASE(Omega(n^1), ?)
323.67/291.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
323.67/291.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
323.67/291.63	
323.67/291.63	
323.67/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
323.67/291.63	
323.67/291.63	(0) CpxTRS
323.67/291.63	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
323.67/291.63	(2) TRS for Loop Detection
323.67/291.63	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
323.67/291.63	(4) BEST
323.67/291.63	    (5) proven lower bound
323.67/291.63	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
323.67/291.63	        (7) BOUNDS(n^1, INF)
323.67/291.63	    (8) TRS for Loop Detection
323.67/291.63	
323.67/291.63	
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(0)
323.67/291.63	Obligation:
323.67/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
323.67/291.63	
323.67/291.63	
323.67/291.63	The TRS R consists of the following rules:
323.67/291.63	
323.67/291.63	   a__zeros -> cons(0, zeros)
323.67/291.63	   a__U11(tt, L) -> s(a__length(mark(L)))
323.67/291.63	   a__and(tt, X) -> mark(X)
323.67/291.63	   a__isNat(0) -> tt
323.67/291.63	   a__isNat(length(V1)) -> a__isNatList(V1)
323.67/291.63	   a__isNat(s(V1)) -> a__isNat(V1)
323.67/291.63	   a__isNatIList(V) -> a__isNatList(V)
323.67/291.63	   a__isNatIList(zeros) -> tt
323.67/291.63	   a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2))
323.67/291.63	   a__isNatList(nil) -> tt
323.67/291.63	   a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2))
323.67/291.63	   a__length(nil) -> 0
323.67/291.63	   a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L)
323.67/291.63	   mark(zeros) -> a__zeros
323.67/291.63	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
323.67/291.63	   mark(length(X)) -> a__length(mark(X))
323.67/291.63	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
323.67/291.63	   mark(isNat(X)) -> a__isNat(X)
323.67/291.63	   mark(isNatList(X)) -> a__isNatList(X)
323.67/291.63	   mark(isNatIList(X)) -> a__isNatIList(X)
323.67/291.63	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
323.67/291.63	   mark(0) -> 0
323.67/291.63	   mark(tt) -> tt
323.67/291.63	   mark(s(X)) -> s(mark(X))
323.67/291.63	   mark(nil) -> nil
323.67/291.63	   a__zeros -> zeros
323.67/291.63	   a__U11(X1, X2) -> U11(X1, X2)
323.67/291.63	   a__length(X) -> length(X)
323.67/291.63	   a__and(X1, X2) -> and(X1, X2)
323.67/291.63	   a__isNat(X) -> isNat(X)
323.67/291.63	   a__isNatList(X) -> isNatList(X)
323.67/291.63	   a__isNatIList(X) -> isNatIList(X)
323.67/291.63	
323.67/291.63	S is empty.
323.67/291.63	Rewrite Strategy: FULL
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
323.67/291.63	Transformed a relative TRS into a decreasing-loop problem.
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(2)
323.67/291.63	Obligation:
323.67/291.63	Analyzing the following TRS for decreasing loops:
323.67/291.63	
323.67/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
323.67/291.63	
323.67/291.63	
323.67/291.63	The TRS R consists of the following rules:
323.67/291.63	
323.67/291.63	   a__zeros -> cons(0, zeros)
323.67/291.63	   a__U11(tt, L) -> s(a__length(mark(L)))
323.67/291.63	   a__and(tt, X) -> mark(X)
323.67/291.63	   a__isNat(0) -> tt
323.67/291.63	   a__isNat(length(V1)) -> a__isNatList(V1)
323.67/291.63	   a__isNat(s(V1)) -> a__isNat(V1)
323.67/291.63	   a__isNatIList(V) -> a__isNatList(V)
323.67/291.63	   a__isNatIList(zeros) -> tt
323.67/291.63	   a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2))
323.67/291.63	   a__isNatList(nil) -> tt
323.67/291.63	   a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2))
323.67/291.63	   a__length(nil) -> 0
323.67/291.63	   a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L)
323.67/291.63	   mark(zeros) -> a__zeros
323.67/291.63	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
323.67/291.63	   mark(length(X)) -> a__length(mark(X))
323.67/291.63	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
323.67/291.63	   mark(isNat(X)) -> a__isNat(X)
323.67/291.63	   mark(isNatList(X)) -> a__isNatList(X)
323.67/291.63	   mark(isNatIList(X)) -> a__isNatIList(X)
323.67/291.63	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
323.67/291.63	   mark(0) -> 0
323.67/291.63	   mark(tt) -> tt
323.67/291.63	   mark(s(X)) -> s(mark(X))
323.67/291.63	   mark(nil) -> nil
323.67/291.63	   a__zeros -> zeros
323.67/291.63	   a__U11(X1, X2) -> U11(X1, X2)
323.67/291.63	   a__length(X) -> length(X)
323.67/291.63	   a__and(X1, X2) -> and(X1, X2)
323.67/291.63	   a__isNat(X) -> isNat(X)
323.67/291.63	   a__isNatList(X) -> isNatList(X)
323.67/291.63	   a__isNatIList(X) -> isNatIList(X)
323.67/291.63	
323.67/291.63	S is empty.
323.67/291.63	Rewrite Strategy: FULL
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(3) DecreasingLoopProof (LOWER BOUND(ID))
323.67/291.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
323.67/291.63	
323.67/291.63	The rewrite sequence
323.67/291.63	
323.67/291.63	mark(length(X)) ->^+ a__length(mark(X))
323.67/291.63	
323.67/291.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
323.67/291.63	
323.67/291.63	The pumping substitution is [X / length(X)].
323.67/291.63	
323.67/291.63	The result substitution is [ ].
323.67/291.63	
323.67/291.63	
323.67/291.63	
323.67/291.63	
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(4)
323.67/291.63	Complex Obligation (BEST)
323.67/291.63	
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(5)
323.67/291.63	Obligation:
323.67/291.63	Proved the lower bound n^1 for the following obligation:
323.67/291.63	
323.67/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
323.67/291.63	
323.67/291.63	
323.67/291.63	The TRS R consists of the following rules:
323.67/291.63	
323.67/291.63	   a__zeros -> cons(0, zeros)
323.67/291.63	   a__U11(tt, L) -> s(a__length(mark(L)))
323.67/291.63	   a__and(tt, X) -> mark(X)
323.67/291.63	   a__isNat(0) -> tt
323.67/291.63	   a__isNat(length(V1)) -> a__isNatList(V1)
323.67/291.63	   a__isNat(s(V1)) -> a__isNat(V1)
323.67/291.63	   a__isNatIList(V) -> a__isNatList(V)
323.67/291.63	   a__isNatIList(zeros) -> tt
323.67/291.63	   a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2))
323.67/291.63	   a__isNatList(nil) -> tt
323.67/291.63	   a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2))
323.67/291.63	   a__length(nil) -> 0
323.67/291.63	   a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L)
323.67/291.63	   mark(zeros) -> a__zeros
323.67/291.63	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
323.67/291.63	   mark(length(X)) -> a__length(mark(X))
323.67/291.63	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
323.67/291.63	   mark(isNat(X)) -> a__isNat(X)
323.67/291.63	   mark(isNatList(X)) -> a__isNatList(X)
323.67/291.63	   mark(isNatIList(X)) -> a__isNatIList(X)
323.67/291.63	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
323.67/291.63	   mark(0) -> 0
323.67/291.63	   mark(tt) -> tt
323.67/291.63	   mark(s(X)) -> s(mark(X))
323.67/291.63	   mark(nil) -> nil
323.67/291.63	   a__zeros -> zeros
323.67/291.63	   a__U11(X1, X2) -> U11(X1, X2)
323.67/291.63	   a__length(X) -> length(X)
323.67/291.63	   a__and(X1, X2) -> and(X1, X2)
323.67/291.63	   a__isNat(X) -> isNat(X)
323.67/291.63	   a__isNatList(X) -> isNatList(X)
323.67/291.63	   a__isNatIList(X) -> isNatIList(X)
323.67/291.63	
323.67/291.63	S is empty.
323.67/291.63	Rewrite Strategy: FULL
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(6) LowerBoundPropagationProof (FINISHED)
323.67/291.63	Propagated lower bound.
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(7)
323.67/291.63	BOUNDS(n^1, INF)
323.67/291.63	
323.67/291.63	----------------------------------------
323.67/291.63	
323.67/291.63	(8)
323.67/291.63	Obligation:
323.67/291.63	Analyzing the following TRS for decreasing loops:
323.67/291.63	
323.67/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
323.67/291.63	
323.67/291.63	
323.67/291.63	The TRS R consists of the following rules:
323.67/291.63	
323.67/291.63	   a__zeros -> cons(0, zeros)
323.67/291.63	   a__U11(tt, L) -> s(a__length(mark(L)))
323.67/291.63	   a__and(tt, X) -> mark(X)
323.67/291.63	   a__isNat(0) -> tt
323.67/291.63	   a__isNat(length(V1)) -> a__isNatList(V1)
323.67/291.63	   a__isNat(s(V1)) -> a__isNat(V1)
323.67/291.63	   a__isNatIList(V) -> a__isNatList(V)
323.67/291.63	   a__isNatIList(zeros) -> tt
323.67/291.63	   a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2))
323.67/291.63	   a__isNatList(nil) -> tt
323.67/291.63	   a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2))
323.67/291.63	   a__length(nil) -> 0
323.67/291.63	   a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L)
323.67/291.63	   mark(zeros) -> a__zeros
323.67/291.63	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
323.67/291.63	   mark(length(X)) -> a__length(mark(X))
323.67/291.63	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
323.67/291.63	   mark(isNat(X)) -> a__isNat(X)
323.67/291.63	   mark(isNatList(X)) -> a__isNatList(X)
323.67/291.63	   mark(isNatIList(X)) -> a__isNatIList(X)
323.67/291.63	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
323.67/291.63	   mark(0) -> 0
323.67/291.63	   mark(tt) -> tt
323.67/291.63	   mark(s(X)) -> s(mark(X))
323.67/291.63	   mark(nil) -> nil
323.67/291.63	   a__zeros -> zeros
323.67/291.63	   a__U11(X1, X2) -> U11(X1, X2)
323.67/291.63	   a__length(X) -> length(X)
323.67/291.63	   a__and(X1, X2) -> and(X1, X2)
323.67/291.63	   a__isNat(X) -> isNat(X)
323.67/291.63	   a__isNatList(X) -> isNatList(X)
323.67/291.63	   a__isNatIList(X) -> isNatIList(X)
323.67/291.63	
323.67/291.63	S is empty.
323.67/291.63	Rewrite Strategy: FULL
323.71/291.66	EOF