10.26/3.43	WORST_CASE(NON_POLY, ?)
10.26/3.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.26/3.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.26/3.45	
10.26/3.45	
10.26/3.45	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
10.26/3.45	
10.26/3.45	(0) CpxTRS
10.26/3.45	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
10.26/3.45	(2) TRS for Loop Detection
10.26/3.45	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
10.26/3.45	(4) BEST
10.26/3.45	    (5) proven lower bound
10.26/3.45	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
10.26/3.45	        (7) BOUNDS(n^1, INF)
10.26/3.45	    (8) TRS for Loop Detection
10.26/3.45	        (9) InfiniteLowerBoundProof [FINISHED, 1265 ms]
10.26/3.45	        (10) BOUNDS(INF, INF)
10.26/3.45	
10.26/3.45	
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(0)
10.26/3.45	Obligation:
10.26/3.45	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
10.26/3.45	
10.26/3.45	
10.26/3.45	The TRS R consists of the following rules:
10.26/3.45	
10.26/3.45	   incr(nil) -> nil
10.26/3.45	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
10.26/3.45	   adx(nil) -> nil
10.26/3.45	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
10.26/3.45	   nats -> adx(zeros)
10.26/3.45	   zeros -> cons(0, n__zeros)
10.26/3.45	   head(cons(X, L)) -> X
10.26/3.45	   tail(cons(X, L)) -> activate(L)
10.26/3.45	   incr(X) -> n__incr(X)
10.26/3.45	   adx(X) -> n__adx(X)
10.26/3.45	   zeros -> n__zeros
10.26/3.45	   activate(n__incr(X)) -> incr(activate(X))
10.26/3.45	   activate(n__adx(X)) -> adx(activate(X))
10.26/3.45	   activate(n__zeros) -> zeros
10.26/3.45	   activate(X) -> X
10.26/3.45	
10.26/3.45	S is empty.
10.26/3.45	Rewrite Strategy: FULL
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
10.26/3.45	Transformed a relative TRS into a decreasing-loop problem.
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(2)
10.26/3.45	Obligation:
10.26/3.45	Analyzing the following TRS for decreasing loops:
10.26/3.45	
10.26/3.45	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
10.26/3.45	
10.26/3.45	
10.26/3.45	The TRS R consists of the following rules:
10.26/3.45	
10.26/3.45	   incr(nil) -> nil
10.26/3.45	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
10.26/3.45	   adx(nil) -> nil
10.26/3.45	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
10.26/3.45	   nats -> adx(zeros)
10.26/3.45	   zeros -> cons(0, n__zeros)
10.26/3.45	   head(cons(X, L)) -> X
10.26/3.45	   tail(cons(X, L)) -> activate(L)
10.26/3.45	   incr(X) -> n__incr(X)
10.26/3.45	   adx(X) -> n__adx(X)
10.26/3.45	   zeros -> n__zeros
10.26/3.45	   activate(n__incr(X)) -> incr(activate(X))
10.26/3.45	   activate(n__adx(X)) -> adx(activate(X))
10.26/3.45	   activate(n__zeros) -> zeros
10.26/3.45	   activate(X) -> X
10.26/3.45	
10.26/3.45	S is empty.
10.26/3.45	Rewrite Strategy: FULL
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(3) DecreasingLoopProof (LOWER BOUND(ID))
10.26/3.45	The following loop(s) give(s) rise to the lower bound Omega(n^1):
10.26/3.45	
10.26/3.45	The rewrite sequence
10.26/3.45	
10.26/3.45	activate(n__adx(X)) ->^+ adx(activate(X))
10.26/3.45	
10.26/3.45	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
10.26/3.45	
10.26/3.45	The pumping substitution is [X / n__adx(X)].
10.26/3.45	
10.26/3.45	The result substitution is [ ].
10.26/3.45	
10.26/3.45	
10.26/3.45	
10.26/3.45	
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(4)
10.26/3.45	Complex Obligation (BEST)
10.26/3.45	
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(5)
10.26/3.45	Obligation:
10.26/3.45	Proved the lower bound n^1 for the following obligation:
10.26/3.45	
10.26/3.45	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
10.26/3.45	
10.26/3.45	
10.26/3.45	The TRS R consists of the following rules:
10.26/3.45	
10.26/3.45	   incr(nil) -> nil
10.26/3.45	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
10.26/3.45	   adx(nil) -> nil
10.26/3.45	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
10.26/3.45	   nats -> adx(zeros)
10.26/3.45	   zeros -> cons(0, n__zeros)
10.26/3.45	   head(cons(X, L)) -> X
10.26/3.45	   tail(cons(X, L)) -> activate(L)
10.26/3.45	   incr(X) -> n__incr(X)
10.26/3.45	   adx(X) -> n__adx(X)
10.26/3.45	   zeros -> n__zeros
10.26/3.45	   activate(n__incr(X)) -> incr(activate(X))
10.26/3.45	   activate(n__adx(X)) -> adx(activate(X))
10.26/3.45	   activate(n__zeros) -> zeros
10.26/3.45	   activate(X) -> X
10.26/3.45	
10.26/3.45	S is empty.
10.26/3.45	Rewrite Strategy: FULL
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(6) LowerBoundPropagationProof (FINISHED)
10.26/3.45	Propagated lower bound.
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(7)
10.26/3.45	BOUNDS(n^1, INF)
10.26/3.45	
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(8)
10.26/3.45	Obligation:
10.26/3.45	Analyzing the following TRS for decreasing loops:
10.26/3.45	
10.26/3.45	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
10.26/3.45	
10.26/3.45	
10.26/3.45	The TRS R consists of the following rules:
10.26/3.45	
10.26/3.45	   incr(nil) -> nil
10.26/3.45	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
10.26/3.45	   adx(nil) -> nil
10.26/3.45	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
10.26/3.45	   nats -> adx(zeros)
10.26/3.45	   zeros -> cons(0, n__zeros)
10.26/3.45	   head(cons(X, L)) -> X
10.26/3.45	   tail(cons(X, L)) -> activate(L)
10.26/3.45	   incr(X) -> n__incr(X)
10.26/3.45	   adx(X) -> n__adx(X)
10.26/3.45	   zeros -> n__zeros
10.26/3.45	   activate(n__incr(X)) -> incr(activate(X))
10.26/3.45	   activate(n__adx(X)) -> adx(activate(X))
10.26/3.45	   activate(n__zeros) -> zeros
10.26/3.45	   activate(X) -> X
10.26/3.45	
10.26/3.45	S is empty.
10.26/3.45	Rewrite Strategy: FULL
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(9) InfiniteLowerBoundProof (FINISHED)
10.26/3.45	The following loop proves infinite runtime complexity:
10.26/3.45	
10.26/3.45	The rewrite sequence
10.26/3.45	
10.26/3.45	activate(n__adx(cons(X1_0, n__zeros))) ->^+ cons(s(X1_0), n__incr(activate(n__adx(cons(0, n__zeros)))))
10.26/3.45	
10.26/3.45	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
10.26/3.45	
10.26/3.45	The pumping substitution is [ ].
10.26/3.45	
10.26/3.45	The result substitution is [X1_0 / 0].
10.26/3.45	
10.26/3.45	
10.26/3.45	
10.26/3.45	
10.26/3.45	----------------------------------------
10.26/3.45	
10.26/3.45	(10)
10.26/3.45	BOUNDS(INF, INF)
10.57/3.49	EOF