17.60/5.36	WORST_CASE(NON_POLY, ?)
17.60/5.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.60/5.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.60/5.37	
17.60/5.37	
17.60/5.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
17.60/5.37	
17.60/5.37	(0) CpxTRS
17.60/5.37	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
17.60/5.37	(2) TRS for Loop Detection
17.60/5.37	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
17.60/5.37	(4) BEST
17.60/5.37	    (5) proven lower bound
17.60/5.37	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
17.60/5.37	        (7) BOUNDS(n^1, INF)
17.60/5.37	    (8) TRS for Loop Detection
17.60/5.37	        (9) InfiniteLowerBoundProof [FINISHED, 2241 ms]
17.60/5.37	        (10) BOUNDS(INF, INF)
17.60/5.37	
17.60/5.37	
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(0)
17.60/5.37	Obligation:
17.60/5.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
17.60/5.37	
17.60/5.37	
17.60/5.37	The TRS R consists of the following rules:
17.60/5.37	
17.60/5.37	   nats -> adx(zeros)
17.60/5.37	   zeros -> cons(n__0, n__zeros)
17.60/5.37	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
17.60/5.37	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
17.60/5.37	   hd(cons(X, Y)) -> activate(X)
17.60/5.37	   tl(cons(X, Y)) -> activate(Y)
17.60/5.37	   0 -> n__0
17.60/5.37	   zeros -> n__zeros
17.60/5.37	   s(X) -> n__s(X)
17.60/5.37	   incr(X) -> n__incr(X)
17.60/5.37	   adx(X) -> n__adx(X)
17.60/5.37	   activate(n__0) -> 0
17.60/5.37	   activate(n__zeros) -> zeros
17.60/5.37	   activate(n__s(X)) -> s(X)
17.60/5.37	   activate(n__incr(X)) -> incr(activate(X))
17.60/5.37	   activate(n__adx(X)) -> adx(activate(X))
17.60/5.37	   activate(X) -> X
17.60/5.37	
17.60/5.37	S is empty.
17.60/5.37	Rewrite Strategy: INNERMOST
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
17.60/5.37	Transformed a relative TRS into a decreasing-loop problem.
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(2)
17.60/5.37	Obligation:
17.60/5.37	Analyzing the following TRS for decreasing loops:
17.60/5.37	
17.60/5.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
17.60/5.37	
17.60/5.37	
17.60/5.37	The TRS R consists of the following rules:
17.60/5.37	
17.60/5.37	   nats -> adx(zeros)
17.60/5.37	   zeros -> cons(n__0, n__zeros)
17.60/5.37	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
17.60/5.37	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
17.60/5.37	   hd(cons(X, Y)) -> activate(X)
17.60/5.37	   tl(cons(X, Y)) -> activate(Y)
17.60/5.37	   0 -> n__0
17.60/5.37	   zeros -> n__zeros
17.60/5.37	   s(X) -> n__s(X)
17.60/5.37	   incr(X) -> n__incr(X)
17.60/5.37	   adx(X) -> n__adx(X)
17.60/5.37	   activate(n__0) -> 0
17.60/5.37	   activate(n__zeros) -> zeros
17.60/5.37	   activate(n__s(X)) -> s(X)
17.60/5.37	   activate(n__incr(X)) -> incr(activate(X))
17.60/5.37	   activate(n__adx(X)) -> adx(activate(X))
17.60/5.37	   activate(X) -> X
17.60/5.37	
17.60/5.37	S is empty.
17.60/5.37	Rewrite Strategy: INNERMOST
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(3) DecreasingLoopProof (LOWER BOUND(ID))
17.60/5.37	The following loop(s) give(s) rise to the lower bound Omega(n^1):
17.60/5.37	
17.60/5.37	The rewrite sequence
17.60/5.37	
17.60/5.37	activate(n__adx(X)) ->^+ adx(activate(X))
17.60/5.37	
17.60/5.37	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
17.60/5.37	
17.60/5.37	The pumping substitution is [X / n__adx(X)].
17.60/5.37	
17.60/5.37	The result substitution is [ ].
17.60/5.37	
17.60/5.37	
17.60/5.37	
17.60/5.37	
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(4)
17.60/5.37	Complex Obligation (BEST)
17.60/5.37	
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(5)
17.60/5.37	Obligation:
17.60/5.37	Proved the lower bound n^1 for the following obligation:
17.60/5.37	
17.60/5.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
17.60/5.37	
17.60/5.37	
17.60/5.37	The TRS R consists of the following rules:
17.60/5.37	
17.60/5.37	   nats -> adx(zeros)
17.60/5.37	   zeros -> cons(n__0, n__zeros)
17.60/5.37	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
17.60/5.37	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
17.60/5.37	   hd(cons(X, Y)) -> activate(X)
17.60/5.37	   tl(cons(X, Y)) -> activate(Y)
17.60/5.37	   0 -> n__0
17.60/5.37	   zeros -> n__zeros
17.60/5.37	   s(X) -> n__s(X)
17.60/5.37	   incr(X) -> n__incr(X)
17.60/5.37	   adx(X) -> n__adx(X)
17.60/5.37	   activate(n__0) -> 0
17.60/5.37	   activate(n__zeros) -> zeros
17.60/5.37	   activate(n__s(X)) -> s(X)
17.60/5.37	   activate(n__incr(X)) -> incr(activate(X))
17.60/5.37	   activate(n__adx(X)) -> adx(activate(X))
17.60/5.37	   activate(X) -> X
17.60/5.37	
17.60/5.37	S is empty.
17.60/5.37	Rewrite Strategy: INNERMOST
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(6) LowerBoundPropagationProof (FINISHED)
17.60/5.37	Propagated lower bound.
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(7)
17.60/5.37	BOUNDS(n^1, INF)
17.60/5.37	
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(8)
17.60/5.37	Obligation:
17.60/5.37	Analyzing the following TRS for decreasing loops:
17.60/5.37	
17.60/5.37	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
17.60/5.37	
17.60/5.37	
17.60/5.37	The TRS R consists of the following rules:
17.60/5.37	
17.60/5.37	   nats -> adx(zeros)
17.60/5.37	   zeros -> cons(n__0, n__zeros)
17.60/5.37	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
17.60/5.37	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
17.60/5.37	   hd(cons(X, Y)) -> activate(X)
17.60/5.37	   tl(cons(X, Y)) -> activate(Y)
17.60/5.37	   0 -> n__0
17.60/5.37	   zeros -> n__zeros
17.60/5.37	   s(X) -> n__s(X)
17.60/5.37	   incr(X) -> n__incr(X)
17.60/5.37	   adx(X) -> n__adx(X)
17.60/5.37	   activate(n__0) -> 0
17.60/5.37	   activate(n__zeros) -> zeros
17.60/5.37	   activate(n__s(X)) -> s(X)
17.60/5.37	   activate(n__incr(X)) -> incr(activate(X))
17.60/5.37	   activate(n__adx(X)) -> adx(activate(X))
17.60/5.37	   activate(X) -> X
17.60/5.37	
17.60/5.37	S is empty.
17.60/5.37	Rewrite Strategy: INNERMOST
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(9) InfiniteLowerBoundProof (FINISHED)
17.60/5.37	The following loop proves infinite runtime complexity:
17.60/5.37	
17.60/5.37	The rewrite sequence
17.60/5.37	
17.60/5.37	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))))))
17.60/5.37	
17.60/5.37	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
17.60/5.37	
17.60/5.37	The pumping substitution is [ ].
17.60/5.37	
17.60/5.37	The result substitution is [X / activate(X1_1), X1_1 / n__0].
17.60/5.37	
17.60/5.37	
17.60/5.37	
17.60/5.37	
17.60/5.37	----------------------------------------
17.60/5.37	
17.60/5.37	(10)
17.60/5.37	BOUNDS(INF, INF)
17.60/5.43	EOF