5.47/2.22	WORST_CASE(NON_POLY, ?)
5.47/2.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.47/2.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.47/2.23	
5.47/2.23	
5.47/2.23	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.47/2.23	
5.47/2.23	(0) CpxTRS
5.47/2.23	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.47/2.23	(2) TRS for Loop Detection
5.47/2.23	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.47/2.23	(4) BEST
5.47/2.23	    (5) proven lower bound
5.47/2.23	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
5.47/2.23	        (7) BOUNDS(n^1, INF)
5.47/2.23	    (8) TRS for Loop Detection
5.47/2.23	        (9) InfiniteLowerBoundProof [FINISHED, 324 ms]
5.47/2.23	        (10) BOUNDS(INF, INF)
5.47/2.23	
5.47/2.23	
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(0)
5.47/2.23	Obligation:
5.47/2.23	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.47/2.23	
5.47/2.23	
5.47/2.23	The TRS R consists of the following rules:
5.47/2.23	
5.47/2.23	   incr(nil) -> nil
5.47/2.23	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
5.47/2.23	   adx(nil) -> nil
5.47/2.23	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
5.47/2.23	   nats -> adx(zeros)
5.47/2.23	   zeros -> cons(0, n__zeros)
5.47/2.23	   head(cons(X, L)) -> X
5.47/2.23	   tail(cons(X, L)) -> activate(L)
5.47/2.23	   incr(X) -> n__incr(X)
5.47/2.23	   adx(X) -> n__adx(X)
5.47/2.23	   zeros -> n__zeros
5.47/2.23	   activate(n__incr(X)) -> incr(activate(X))
5.47/2.23	   activate(n__adx(X)) -> adx(activate(X))
5.47/2.23	   activate(n__zeros) -> zeros
5.47/2.23	   activate(X) -> X
5.47/2.23	
5.47/2.23	S is empty.
5.47/2.23	Rewrite Strategy: INNERMOST
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.47/2.23	Transformed a relative TRS into a decreasing-loop problem.
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(2)
5.47/2.23	Obligation:
5.47/2.23	Analyzing the following TRS for decreasing loops:
5.47/2.23	
5.47/2.23	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.47/2.23	
5.47/2.23	
5.47/2.23	The TRS R consists of the following rules:
5.47/2.23	
5.47/2.23	   incr(nil) -> nil
5.47/2.23	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
5.47/2.23	   adx(nil) -> nil
5.47/2.23	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
5.47/2.23	   nats -> adx(zeros)
5.47/2.23	   zeros -> cons(0, n__zeros)
5.47/2.23	   head(cons(X, L)) -> X
5.47/2.23	   tail(cons(X, L)) -> activate(L)
5.47/2.23	   incr(X) -> n__incr(X)
5.47/2.23	   adx(X) -> n__adx(X)
5.47/2.23	   zeros -> n__zeros
5.47/2.23	   activate(n__incr(X)) -> incr(activate(X))
5.47/2.23	   activate(n__adx(X)) -> adx(activate(X))
5.47/2.23	   activate(n__zeros) -> zeros
5.47/2.23	   activate(X) -> X
5.47/2.23	
5.47/2.23	S is empty.
5.47/2.23	Rewrite Strategy: INNERMOST
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(3) DecreasingLoopProof (LOWER BOUND(ID))
5.47/2.23	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.47/2.23	
5.47/2.23	The rewrite sequence
5.47/2.23	
5.47/2.23	activate(n__adx(X)) ->^+ adx(activate(X))
5.47/2.23	
5.47/2.23	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.47/2.23	
5.47/2.23	The pumping substitution is [X / n__adx(X)].
5.47/2.23	
5.47/2.23	The result substitution is [ ].
5.47/2.23	
5.47/2.23	
5.47/2.23	
5.47/2.23	
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(4)
5.47/2.23	Complex Obligation (BEST)
5.47/2.23	
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(5)
5.47/2.23	Obligation:
5.47/2.23	Proved the lower bound n^1 for the following obligation:
5.47/2.23	
5.47/2.23	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.47/2.23	
5.47/2.23	
5.47/2.23	The TRS R consists of the following rules:
5.47/2.23	
5.47/2.23	   incr(nil) -> nil
5.47/2.23	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
5.47/2.23	   adx(nil) -> nil
5.47/2.23	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
5.47/2.23	   nats -> adx(zeros)
5.47/2.23	   zeros -> cons(0, n__zeros)
5.47/2.23	   head(cons(X, L)) -> X
5.47/2.23	   tail(cons(X, L)) -> activate(L)
5.47/2.23	   incr(X) -> n__incr(X)
5.47/2.23	   adx(X) -> n__adx(X)
5.47/2.23	   zeros -> n__zeros
5.47/2.23	   activate(n__incr(X)) -> incr(activate(X))
5.47/2.23	   activate(n__adx(X)) -> adx(activate(X))
5.47/2.23	   activate(n__zeros) -> zeros
5.47/2.23	   activate(X) -> X
5.47/2.23	
5.47/2.23	S is empty.
5.47/2.23	Rewrite Strategy: INNERMOST
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(6) LowerBoundPropagationProof (FINISHED)
5.47/2.23	Propagated lower bound.
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(7)
5.47/2.23	BOUNDS(n^1, INF)
5.47/2.23	
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(8)
5.47/2.23	Obligation:
5.47/2.23	Analyzing the following TRS for decreasing loops:
5.47/2.23	
5.47/2.23	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.47/2.23	
5.47/2.23	
5.47/2.23	The TRS R consists of the following rules:
5.47/2.23	
5.47/2.23	   incr(nil) -> nil
5.47/2.23	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
5.47/2.23	   adx(nil) -> nil
5.47/2.23	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
5.47/2.23	   nats -> adx(zeros)
5.47/2.23	   zeros -> cons(0, n__zeros)
5.47/2.23	   head(cons(X, L)) -> X
5.47/2.23	   tail(cons(X, L)) -> activate(L)
5.47/2.23	   incr(X) -> n__incr(X)
5.47/2.23	   adx(X) -> n__adx(X)
5.47/2.23	   zeros -> n__zeros
5.47/2.23	   activate(n__incr(X)) -> incr(activate(X))
5.47/2.23	   activate(n__adx(X)) -> adx(activate(X))
5.47/2.23	   activate(n__zeros) -> zeros
5.47/2.23	   activate(X) -> X
5.47/2.23	
5.47/2.23	S is empty.
5.47/2.23	Rewrite Strategy: INNERMOST
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(9) InfiniteLowerBoundProof (FINISHED)
5.47/2.23	The following loop proves infinite runtime complexity:
5.47/2.23	
5.47/2.23	The rewrite sequence
5.47/2.23	
5.47/2.23	activate(n__adx(cons(X1_0, n__zeros))) ->^+ cons(s(X1_0), n__incr(activate(n__adx(cons(0, n__zeros)))))
5.47/2.23	
5.47/2.23	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
5.47/2.23	
5.47/2.23	The pumping substitution is [ ].
5.47/2.23	
5.47/2.23	The result substitution is [X1_0 / 0].
5.47/2.23	
5.47/2.23	
5.47/2.23	
5.47/2.23	
5.47/2.23	----------------------------------------
5.47/2.23	
5.47/2.23	(10)
5.47/2.23	BOUNDS(INF, INF)
5.61/2.27	EOF