33.63/10.23	WORST_CASE(NON_POLY, ?)
33.94/10.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.94/10.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.94/10.23	
33.94/10.23	
33.94/10.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
33.94/10.23	
33.94/10.23	(0) CpxTRS
33.94/10.23	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
33.94/10.23	(2) TRS for Loop Detection
33.94/10.23	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
33.94/10.23	(4) BEST
33.94/10.23	    (5) proven lower bound
33.94/10.23	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
33.94/10.23	        (7) BOUNDS(n^1, INF)
33.94/10.23	    (8) TRS for Loop Detection
33.94/10.23	        (9) InfiniteLowerBoundProof [FINISHED, 7074 ms]
33.94/10.23	        (10) BOUNDS(INF, INF)
33.94/10.23	
33.94/10.23	
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(0)
33.94/10.23	Obligation:
33.94/10.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
33.94/10.23	
33.94/10.23	
33.94/10.23	The TRS R consists of the following rules:
33.94/10.23	
33.94/10.23	   nats -> adx(zeros)
33.94/10.23	   zeros -> cons(n__0, n__zeros)
33.94/10.23	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
33.94/10.23	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
33.94/10.23	   hd(cons(X, Y)) -> activate(X)
33.94/10.23	   tl(cons(X, Y)) -> activate(Y)
33.94/10.23	   0 -> n__0
33.94/10.23	   zeros -> n__zeros
33.94/10.23	   s(X) -> n__s(X)
33.94/10.23	   incr(X) -> n__incr(X)
33.94/10.23	   adx(X) -> n__adx(X)
33.94/10.23	   activate(n__0) -> 0
33.94/10.23	   activate(n__zeros) -> zeros
33.94/10.23	   activate(n__s(X)) -> s(X)
33.94/10.23	   activate(n__incr(X)) -> incr(activate(X))
33.94/10.23	   activate(n__adx(X)) -> adx(activate(X))
33.94/10.23	   activate(X) -> X
33.94/10.23	
33.94/10.23	S is empty.
33.94/10.23	Rewrite Strategy: FULL
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
33.94/10.23	Transformed a relative TRS into a decreasing-loop problem.
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(2)
33.94/10.23	Obligation:
33.94/10.23	Analyzing the following TRS for decreasing loops:
33.94/10.23	
33.94/10.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
33.94/10.23	
33.94/10.23	
33.94/10.23	The TRS R consists of the following rules:
33.94/10.23	
33.94/10.23	   nats -> adx(zeros)
33.94/10.23	   zeros -> cons(n__0, n__zeros)
33.94/10.23	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
33.94/10.23	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
33.94/10.23	   hd(cons(X, Y)) -> activate(X)
33.94/10.23	   tl(cons(X, Y)) -> activate(Y)
33.94/10.23	   0 -> n__0
33.94/10.23	   zeros -> n__zeros
33.94/10.23	   s(X) -> n__s(X)
33.94/10.23	   incr(X) -> n__incr(X)
33.94/10.23	   adx(X) -> n__adx(X)
33.94/10.23	   activate(n__0) -> 0
33.94/10.23	   activate(n__zeros) -> zeros
33.94/10.23	   activate(n__s(X)) -> s(X)
33.94/10.23	   activate(n__incr(X)) -> incr(activate(X))
33.94/10.23	   activate(n__adx(X)) -> adx(activate(X))
33.94/10.23	   activate(X) -> X
33.94/10.23	
33.94/10.23	S is empty.
33.94/10.23	Rewrite Strategy: FULL
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(3) DecreasingLoopProof (LOWER BOUND(ID))
33.94/10.23	The following loop(s) give(s) rise to the lower bound Omega(n^1):
33.94/10.23	
33.94/10.23	The rewrite sequence
33.94/10.23	
33.94/10.23	activate(n__adx(X)) ->^+ adx(activate(X))
33.94/10.23	
33.94/10.23	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
33.94/10.23	
33.94/10.23	The pumping substitution is [X / n__adx(X)].
33.94/10.23	
33.94/10.23	The result substitution is [ ].
33.94/10.23	
33.94/10.23	
33.94/10.23	
33.94/10.23	
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(4)
33.94/10.23	Complex Obligation (BEST)
33.94/10.23	
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(5)
33.94/10.23	Obligation:
33.94/10.23	Proved the lower bound n^1 for the following obligation:
33.94/10.23	
33.94/10.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
33.94/10.23	
33.94/10.23	
33.94/10.23	The TRS R consists of the following rules:
33.94/10.23	
33.94/10.23	   nats -> adx(zeros)
33.94/10.23	   zeros -> cons(n__0, n__zeros)
33.94/10.23	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
33.94/10.23	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
33.94/10.23	   hd(cons(X, Y)) -> activate(X)
33.94/10.23	   tl(cons(X, Y)) -> activate(Y)
33.94/10.23	   0 -> n__0
33.94/10.23	   zeros -> n__zeros
33.94/10.23	   s(X) -> n__s(X)
33.94/10.23	   incr(X) -> n__incr(X)
33.94/10.23	   adx(X) -> n__adx(X)
33.94/10.23	   activate(n__0) -> 0
33.94/10.23	   activate(n__zeros) -> zeros
33.94/10.23	   activate(n__s(X)) -> s(X)
33.94/10.23	   activate(n__incr(X)) -> incr(activate(X))
33.94/10.23	   activate(n__adx(X)) -> adx(activate(X))
33.94/10.23	   activate(X) -> X
33.94/10.23	
33.94/10.23	S is empty.
33.94/10.23	Rewrite Strategy: FULL
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(6) LowerBoundPropagationProof (FINISHED)
33.94/10.23	Propagated lower bound.
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(7)
33.94/10.23	BOUNDS(n^1, INF)
33.94/10.23	
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(8)
33.94/10.23	Obligation:
33.94/10.23	Analyzing the following TRS for decreasing loops:
33.94/10.23	
33.94/10.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
33.94/10.23	
33.94/10.23	
33.94/10.23	The TRS R consists of the following rules:
33.94/10.23	
33.94/10.23	   nats -> adx(zeros)
33.94/10.23	   zeros -> cons(n__0, n__zeros)
33.94/10.23	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
33.94/10.23	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
33.94/10.23	   hd(cons(X, Y)) -> activate(X)
33.94/10.23	   tl(cons(X, Y)) -> activate(Y)
33.94/10.23	   0 -> n__0
33.94/10.23	   zeros -> n__zeros
33.94/10.23	   s(X) -> n__s(X)
33.94/10.23	   incr(X) -> n__incr(X)
33.94/10.23	   adx(X) -> n__adx(X)
33.94/10.23	   activate(n__0) -> 0
33.94/10.23	   activate(n__zeros) -> zeros
33.94/10.23	   activate(n__s(X)) -> s(X)
33.94/10.23	   activate(n__incr(X)) -> incr(activate(X))
33.94/10.23	   activate(n__adx(X)) -> adx(activate(X))
33.94/10.23	   activate(X) -> X
33.94/10.23	
33.94/10.23	S is empty.
33.94/10.23	Rewrite Strategy: FULL
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(9) InfiniteLowerBoundProof (FINISHED)
33.94/10.23	The following loop proves infinite runtime complexity:
33.94/10.23	
33.94/10.23	The rewrite sequence
33.94/10.23	
33.94/10.23	incr(cons(X, n__adx(cons(X1_1, n__zeros)))) ->^+ cons(n__s(activate(X)), n__incr(incr(cons(activate(X1_1), n__adx(cons(n__0, n__zeros))))))
33.94/10.23	
33.94/10.23	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
33.94/10.23	
33.94/10.23	The pumping substitution is [ ].
33.94/10.23	
33.94/10.23	The result substitution is [X / activate(X1_1), X1_1 / n__0].
33.94/10.23	
33.94/10.23	
33.94/10.23	
33.94/10.23	
33.94/10.23	----------------------------------------
33.94/10.23	
33.94/10.23	(10)
33.94/10.23	BOUNDS(INF, INF)
33.98/10.28	EOF