4.90/2.12	WORST_CASE(NON_POLY, ?)
4.90/2.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.90/2.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.90/2.13	
4.90/2.13	
4.90/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.90/2.13	
4.90/2.13	(0) CpxTRS
4.90/2.13	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.90/2.13	(2) TRS for Loop Detection
4.90/2.13	(3) DecreasingLoopProof [LOWER BOUND(ID), 37 ms]
4.90/2.13	(4) BEST
4.90/2.13	    (5) proven lower bound
4.90/2.13	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
4.90/2.13	        (7) BOUNDS(n^1, INF)
4.90/2.13	    (8) TRS for Loop Detection
4.90/2.13	        (9) InfiniteLowerBoundProof [FINISHED, 250 ms]
4.90/2.13	        (10) BOUNDS(INF, INF)
4.90/2.13	
4.90/2.13	
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(0)
4.90/2.13	Obligation:
4.90/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.90/2.13	
4.90/2.13	
4.90/2.13	The TRS R consists of the following rules:
4.90/2.13	
4.90/2.13	   incr(nil) -> nil
4.90/2.13	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
4.90/2.13	   adx(nil) -> nil
4.90/2.13	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
4.90/2.13	   nats -> adx(zeros)
4.90/2.13	   zeros -> cons(0, n__zeros)
4.90/2.13	   head(cons(X, L)) -> X
4.90/2.13	   tail(cons(X, L)) -> activate(L)
4.90/2.13	   incr(X) -> n__incr(X)
4.90/2.13	   adx(X) -> n__adx(X)
4.90/2.13	   zeros -> n__zeros
4.90/2.13	   activate(n__incr(X)) -> incr(X)
4.90/2.13	   activate(n__adx(X)) -> adx(X)
4.90/2.13	   activate(n__zeros) -> zeros
4.90/2.13	   activate(X) -> X
4.90/2.13	
4.90/2.13	S is empty.
4.90/2.13	Rewrite Strategy: FULL
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.90/2.13	Transformed a relative TRS into a decreasing-loop problem.
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(2)
4.90/2.13	Obligation:
4.90/2.13	Analyzing the following TRS for decreasing loops:
4.90/2.13	
4.90/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.90/2.13	
4.90/2.13	
4.90/2.13	The TRS R consists of the following rules:
4.90/2.13	
4.90/2.13	   incr(nil) -> nil
4.90/2.13	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
4.90/2.13	   adx(nil) -> nil
4.90/2.13	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
4.90/2.13	   nats -> adx(zeros)
4.90/2.13	   zeros -> cons(0, n__zeros)
4.90/2.13	   head(cons(X, L)) -> X
4.90/2.13	   tail(cons(X, L)) -> activate(L)
4.90/2.13	   incr(X) -> n__incr(X)
4.90/2.13	   adx(X) -> n__adx(X)
4.90/2.13	   zeros -> n__zeros
4.90/2.13	   activate(n__incr(X)) -> incr(X)
4.90/2.13	   activate(n__adx(X)) -> adx(X)
4.90/2.13	   activate(n__zeros) -> zeros
4.90/2.13	   activate(X) -> X
4.90/2.13	
4.90/2.13	S is empty.
4.90/2.13	Rewrite Strategy: FULL
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(3) DecreasingLoopProof (LOWER BOUND(ID))
4.90/2.13	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.90/2.13	
4.90/2.13	The rewrite sequence
4.90/2.13	
4.90/2.13	activate(n__incr(cons(X1_0, L2_0))) ->^+ cons(s(X1_0), n__incr(activate(L2_0)))
4.90/2.13	
4.90/2.13	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
4.90/2.13	
4.90/2.13	The pumping substitution is [L2_0 / n__incr(cons(X1_0, L2_0))].
4.90/2.13	
4.90/2.13	The result substitution is [ ].
4.90/2.13	
4.90/2.13	
4.90/2.13	
4.90/2.13	
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(4)
4.90/2.13	Complex Obligation (BEST)
4.90/2.13	
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(5)
4.90/2.13	Obligation:
4.90/2.13	Proved the lower bound n^1 for the following obligation:
4.90/2.13	
4.90/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.90/2.13	
4.90/2.13	
4.90/2.13	The TRS R consists of the following rules:
4.90/2.13	
4.90/2.13	   incr(nil) -> nil
4.90/2.13	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
4.90/2.13	   adx(nil) -> nil
4.90/2.13	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
4.90/2.13	   nats -> adx(zeros)
4.90/2.13	   zeros -> cons(0, n__zeros)
4.90/2.13	   head(cons(X, L)) -> X
4.90/2.13	   tail(cons(X, L)) -> activate(L)
4.90/2.13	   incr(X) -> n__incr(X)
4.90/2.13	   adx(X) -> n__adx(X)
4.90/2.13	   zeros -> n__zeros
4.90/2.13	   activate(n__incr(X)) -> incr(X)
4.90/2.13	   activate(n__adx(X)) -> adx(X)
4.90/2.13	   activate(n__zeros) -> zeros
4.90/2.13	   activate(X) -> X
4.90/2.13	
4.90/2.13	S is empty.
4.90/2.13	Rewrite Strategy: FULL
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(6) LowerBoundPropagationProof (FINISHED)
4.90/2.13	Propagated lower bound.
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(7)
4.90/2.13	BOUNDS(n^1, INF)
4.90/2.13	
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(8)
4.90/2.13	Obligation:
4.90/2.13	Analyzing the following TRS for decreasing loops:
4.90/2.13	
4.90/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
4.90/2.13	
4.90/2.13	
4.90/2.13	The TRS R consists of the following rules:
4.90/2.13	
4.90/2.13	   incr(nil) -> nil
4.90/2.13	   incr(cons(X, L)) -> cons(s(X), n__incr(activate(L)))
4.90/2.13	   adx(nil) -> nil
4.90/2.13	   adx(cons(X, L)) -> incr(cons(X, n__adx(activate(L))))
4.90/2.13	   nats -> adx(zeros)
4.90/2.13	   zeros -> cons(0, n__zeros)
4.90/2.13	   head(cons(X, L)) -> X
4.90/2.13	   tail(cons(X, L)) -> activate(L)
4.90/2.13	   incr(X) -> n__incr(X)
4.90/2.13	   adx(X) -> n__adx(X)
4.90/2.13	   zeros -> n__zeros
4.90/2.13	   activate(n__incr(X)) -> incr(X)
4.90/2.13	   activate(n__adx(X)) -> adx(X)
4.90/2.13	   activate(n__zeros) -> zeros
4.90/2.13	   activate(X) -> X
4.90/2.13	
4.90/2.13	S is empty.
4.90/2.13	Rewrite Strategy: FULL
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(9) InfiniteLowerBoundProof (FINISHED)
4.90/2.13	The following loop proves infinite runtime complexity:
4.90/2.13	
4.90/2.13	The rewrite sequence
4.90/2.13	
4.90/2.13	adx(cons(X, n__zeros)) ->^+ cons(s(X), n__incr(adx(cons(0, n__zeros))))
4.90/2.13	
4.90/2.13	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
4.90/2.13	
4.90/2.13	The pumping substitution is [ ].
4.90/2.13	
4.90/2.13	The result substitution is [X / 0].
4.90/2.13	
4.90/2.13	
4.90/2.13	
4.90/2.13	
4.90/2.13	----------------------------------------
4.90/2.13	
4.90/2.13	(10)
4.90/2.13	BOUNDS(INF, INF)
4.90/2.19	EOF