1105.59/291.50	WORST_CASE(Omega(n^1), ?)
1105.59/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1105.59/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.59/291.53	
1105.59/291.53	(0) CpxTRS
1105.59/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1105.59/291.53	(2) TRS for Loop Detection
1105.59/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1105.59/291.53	(4) BEST
1105.59/291.53	    (5) proven lower bound
1105.59/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1105.59/291.53	        (7) BOUNDS(n^1, INF)
1105.59/291.53	    (8) TRS for Loop Detection
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(0)
1105.59/291.53	Obligation:
1105.59/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	The TRS R consists of the following rules:
1105.59/291.53	
1105.59/291.53	   a__nats -> a__adx(a__zeros)
1105.59/291.53	   a__zeros -> cons(0, zeros)
1105.59/291.53	   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
1105.59/291.53	   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
1105.59/291.53	   a__hd(cons(X, Y)) -> mark(X)
1105.59/291.53	   a__tl(cons(X, Y)) -> mark(Y)
1105.59/291.53	   mark(nats) -> a__nats
1105.59/291.53	   mark(adx(X)) -> a__adx(mark(X))
1105.59/291.53	   mark(zeros) -> a__zeros
1105.59/291.53	   mark(incr(X)) -> a__incr(mark(X))
1105.59/291.53	   mark(hd(X)) -> a__hd(mark(X))
1105.59/291.53	   mark(tl(X)) -> a__tl(mark(X))
1105.59/291.53	   mark(cons(X1, X2)) -> cons(X1, X2)
1105.59/291.53	   mark(0) -> 0
1105.59/291.53	   mark(s(X)) -> s(X)
1105.59/291.53	   a__nats -> nats
1105.59/291.53	   a__adx(X) -> adx(X)
1105.59/291.53	   a__zeros -> zeros
1105.59/291.53	   a__incr(X) -> incr(X)
1105.59/291.53	   a__hd(X) -> hd(X)
1105.59/291.53	   a__tl(X) -> tl(X)
1105.59/291.53	
1105.59/291.53	S is empty.
1105.59/291.53	Rewrite Strategy: FULL
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1105.59/291.53	Transformed a relative TRS into a decreasing-loop problem.
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(2)
1105.59/291.53	Obligation:
1105.59/291.53	Analyzing the following TRS for decreasing loops:
1105.59/291.53	
1105.59/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	The TRS R consists of the following rules:
1105.59/291.53	
1105.59/291.53	   a__nats -> a__adx(a__zeros)
1105.59/291.53	   a__zeros -> cons(0, zeros)
1105.59/291.53	   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
1105.59/291.53	   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
1105.59/291.53	   a__hd(cons(X, Y)) -> mark(X)
1105.59/291.53	   a__tl(cons(X, Y)) -> mark(Y)
1105.59/291.53	   mark(nats) -> a__nats
1105.59/291.53	   mark(adx(X)) -> a__adx(mark(X))
1105.59/291.53	   mark(zeros) -> a__zeros
1105.59/291.53	   mark(incr(X)) -> a__incr(mark(X))
1105.59/291.53	   mark(hd(X)) -> a__hd(mark(X))
1105.59/291.53	   mark(tl(X)) -> a__tl(mark(X))
1105.59/291.53	   mark(cons(X1, X2)) -> cons(X1, X2)
1105.59/291.53	   mark(0) -> 0
1105.59/291.53	   mark(s(X)) -> s(X)
1105.59/291.53	   a__nats -> nats
1105.59/291.53	   a__adx(X) -> adx(X)
1105.59/291.53	   a__zeros -> zeros
1105.59/291.53	   a__incr(X) -> incr(X)
1105.59/291.53	   a__hd(X) -> hd(X)
1105.59/291.53	   a__tl(X) -> tl(X)
1105.59/291.53	
1105.59/291.53	S is empty.
1105.59/291.53	Rewrite Strategy: FULL
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
1105.59/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1105.59/291.53	
1105.59/291.53	The rewrite sequence
1105.59/291.53	
1105.59/291.53	mark(incr(X)) ->^+ a__incr(mark(X))
1105.59/291.53	
1105.59/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1105.59/291.53	
1105.59/291.53	The pumping substitution is [X / incr(X)].
1105.59/291.53	
1105.59/291.53	The result substitution is [ ].
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(4)
1105.59/291.53	Complex Obligation (BEST)
1105.59/291.53	
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(5)
1105.59/291.53	Obligation:
1105.59/291.53	Proved the lower bound n^1 for the following obligation:
1105.59/291.53	
1105.59/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	The TRS R consists of the following rules:
1105.59/291.53	
1105.59/291.53	   a__nats -> a__adx(a__zeros)
1105.59/291.53	   a__zeros -> cons(0, zeros)
1105.59/291.53	   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
1105.59/291.53	   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
1105.59/291.53	   a__hd(cons(X, Y)) -> mark(X)
1105.59/291.53	   a__tl(cons(X, Y)) -> mark(Y)
1105.59/291.53	   mark(nats) -> a__nats
1105.59/291.53	   mark(adx(X)) -> a__adx(mark(X))
1105.59/291.53	   mark(zeros) -> a__zeros
1105.59/291.53	   mark(incr(X)) -> a__incr(mark(X))
1105.59/291.53	   mark(hd(X)) -> a__hd(mark(X))
1105.59/291.53	   mark(tl(X)) -> a__tl(mark(X))
1105.59/291.53	   mark(cons(X1, X2)) -> cons(X1, X2)
1105.59/291.53	   mark(0) -> 0
1105.59/291.53	   mark(s(X)) -> s(X)
1105.59/291.53	   a__nats -> nats
1105.59/291.53	   a__adx(X) -> adx(X)
1105.59/291.53	   a__zeros -> zeros
1105.59/291.53	   a__incr(X) -> incr(X)
1105.59/291.53	   a__hd(X) -> hd(X)
1105.59/291.53	   a__tl(X) -> tl(X)
1105.59/291.53	
1105.59/291.53	S is empty.
1105.59/291.53	Rewrite Strategy: FULL
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(6) LowerBoundPropagationProof (FINISHED)
1105.59/291.53	Propagated lower bound.
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(7)
1105.59/291.53	BOUNDS(n^1, INF)
1105.59/291.53	
1105.59/291.53	----------------------------------------
1105.59/291.53	
1105.59/291.53	(8)
1105.59/291.53	Obligation:
1105.59/291.53	Analyzing the following TRS for decreasing loops:
1105.59/291.53	
1105.59/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.59/291.53	
1105.59/291.53	
1105.59/291.53	The TRS R consists of the following rules:
1105.59/291.53	
1105.59/291.53	   a__nats -> a__adx(a__zeros)
1105.59/291.53	   a__zeros -> cons(0, zeros)
1105.59/291.53	   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
1105.59/291.53	   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
1105.59/291.53	   a__hd(cons(X, Y)) -> mark(X)
1105.59/291.53	   a__tl(cons(X, Y)) -> mark(Y)
1105.59/291.53	   mark(nats) -> a__nats
1105.59/291.53	   mark(adx(X)) -> a__adx(mark(X))
1105.59/291.53	   mark(zeros) -> a__zeros
1105.59/291.53	   mark(incr(X)) -> a__incr(mark(X))
1105.59/291.53	   mark(hd(X)) -> a__hd(mark(X))
1105.59/291.53	   mark(tl(X)) -> a__tl(mark(X))
1105.59/291.53	   mark(cons(X1, X2)) -> cons(X1, X2)
1105.59/291.53	   mark(0) -> 0
1105.59/291.53	   mark(s(X)) -> s(X)
1105.59/291.53	   a__nats -> nats
1105.59/291.53	   a__adx(X) -> adx(X)
1105.59/291.53	   a__zeros -> zeros
1105.59/291.53	   a__incr(X) -> incr(X)
1105.59/291.53	   a__hd(X) -> hd(X)
1105.59/291.53	   a__tl(X) -> tl(X)
1105.59/291.53	
1105.59/291.53	S is empty.
1105.59/291.53	Rewrite Strategy: FULL
1105.80/291.60	EOF