11.12/3.73	WORST_CASE(NON_POLY, ?)
11.12/3.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.12/3.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.12/3.74	
11.12/3.74	
11.12/3.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
11.12/3.74	
11.12/3.74	(0) CpxTRS
11.12/3.74	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
11.12/3.74	(2) TRS for Loop Detection
11.12/3.74	(3) DecreasingLoopProof [LOWER BOUND(ID), 43 ms]
11.12/3.74	(4) BEST
11.12/3.74	    (5) proven lower bound
11.12/3.74	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
11.12/3.74	        (7) BOUNDS(n^1, INF)
11.12/3.74	    (8) TRS for Loop Detection
11.12/3.74	        (9) InfiniteLowerBoundProof [FINISHED, 1361 ms]
11.12/3.74	        (10) BOUNDS(INF, INF)
11.12/3.74	
11.12/3.74	
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(0)
11.12/3.74	Obligation:
11.12/3.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
11.12/3.74	
11.12/3.74	
11.12/3.74	The TRS R consists of the following rules:
11.12/3.74	
11.12/3.74	   nats -> adx(zeros)
11.12/3.74	   zeros -> cons(n__0, n__zeros)
11.12/3.74	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
11.12/3.74	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
11.12/3.74	   hd(cons(X, Y)) -> activate(X)
11.12/3.74	   tl(cons(X, Y)) -> activate(Y)
11.12/3.74	   0 -> n__0
11.12/3.74	   zeros -> n__zeros
11.12/3.74	   s(X) -> n__s(X)
11.12/3.74	   incr(X) -> n__incr(X)
11.12/3.74	   adx(X) -> n__adx(X)
11.12/3.74	   activate(n__0) -> 0
11.12/3.74	   activate(n__zeros) -> zeros
11.12/3.74	   activate(n__s(X)) -> s(X)
11.12/3.74	   activate(n__incr(X)) -> incr(X)
11.12/3.74	   activate(n__adx(X)) -> adx(X)
11.12/3.74	   activate(X) -> X
11.12/3.74	
11.12/3.74	S is empty.
11.12/3.74	Rewrite Strategy: FULL
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
11.12/3.74	Transformed a relative TRS into a decreasing-loop problem.
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(2)
11.12/3.74	Obligation:
11.12/3.74	Analyzing the following TRS for decreasing loops:
11.12/3.74	
11.12/3.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
11.12/3.74	
11.12/3.74	
11.12/3.74	The TRS R consists of the following rules:
11.12/3.74	
11.12/3.74	   nats -> adx(zeros)
11.12/3.74	   zeros -> cons(n__0, n__zeros)
11.12/3.74	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
11.12/3.74	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
11.12/3.74	   hd(cons(X, Y)) -> activate(X)
11.12/3.74	   tl(cons(X, Y)) -> activate(Y)
11.12/3.74	   0 -> n__0
11.12/3.74	   zeros -> n__zeros
11.12/3.74	   s(X) -> n__s(X)
11.12/3.74	   incr(X) -> n__incr(X)
11.12/3.74	   adx(X) -> n__adx(X)
11.12/3.74	   activate(n__0) -> 0
11.12/3.74	   activate(n__zeros) -> zeros
11.12/3.74	   activate(n__s(X)) -> s(X)
11.12/3.74	   activate(n__incr(X)) -> incr(X)
11.12/3.74	   activate(n__adx(X)) -> adx(X)
11.12/3.74	   activate(X) -> X
11.12/3.74	
11.12/3.74	S is empty.
11.12/3.74	Rewrite Strategy: FULL
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(3) DecreasingLoopProof (LOWER BOUND(ID))
11.12/3.74	The following loop(s) give(s) rise to the lower bound Omega(n^1):
11.12/3.74	
11.12/3.74	The rewrite sequence
11.12/3.74	
11.12/3.74	activate(n__incr(cons(X1_0, Y2_0))) ->^+ cons(n__s(activate(X1_0)), n__incr(activate(Y2_0)))
11.12/3.74	
11.12/3.74	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
11.12/3.74	
11.12/3.74	The pumping substitution is [X1_0 / n__incr(cons(X1_0, Y2_0))].
11.12/3.74	
11.12/3.74	The result substitution is [ ].
11.12/3.74	
11.12/3.74	
11.12/3.74	
11.12/3.74	
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(4)
11.12/3.74	Complex Obligation (BEST)
11.12/3.74	
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(5)
11.12/3.74	Obligation:
11.12/3.74	Proved the lower bound n^1 for the following obligation:
11.12/3.74	
11.12/3.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
11.12/3.74	
11.12/3.74	
11.12/3.74	The TRS R consists of the following rules:
11.12/3.74	
11.12/3.74	   nats -> adx(zeros)
11.12/3.74	   zeros -> cons(n__0, n__zeros)
11.12/3.74	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
11.12/3.74	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
11.12/3.74	   hd(cons(X, Y)) -> activate(X)
11.12/3.74	   tl(cons(X, Y)) -> activate(Y)
11.12/3.74	   0 -> n__0
11.12/3.74	   zeros -> n__zeros
11.12/3.74	   s(X) -> n__s(X)
11.12/3.74	   incr(X) -> n__incr(X)
11.12/3.74	   adx(X) -> n__adx(X)
11.12/3.74	   activate(n__0) -> 0
11.12/3.74	   activate(n__zeros) -> zeros
11.12/3.74	   activate(n__s(X)) -> s(X)
11.12/3.74	   activate(n__incr(X)) -> incr(X)
11.12/3.74	   activate(n__adx(X)) -> adx(X)
11.12/3.74	   activate(X) -> X
11.12/3.74	
11.12/3.74	S is empty.
11.12/3.74	Rewrite Strategy: FULL
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(6) LowerBoundPropagationProof (FINISHED)
11.12/3.74	Propagated lower bound.
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(7)
11.12/3.74	BOUNDS(n^1, INF)
11.12/3.74	
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(8)
11.12/3.74	Obligation:
11.12/3.74	Analyzing the following TRS for decreasing loops:
11.12/3.74	
11.12/3.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
11.12/3.74	
11.12/3.74	
11.12/3.74	The TRS R consists of the following rules:
11.12/3.74	
11.12/3.74	   nats -> adx(zeros)
11.12/3.74	   zeros -> cons(n__0, n__zeros)
11.12/3.74	   incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y)))
11.12/3.74	   adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y))))
11.12/3.74	   hd(cons(X, Y)) -> activate(X)
11.12/3.74	   tl(cons(X, Y)) -> activate(Y)
11.12/3.74	   0 -> n__0
11.12/3.74	   zeros -> n__zeros
11.12/3.74	   s(X) -> n__s(X)
11.12/3.74	   incr(X) -> n__incr(X)
11.12/3.74	   adx(X) -> n__adx(X)
11.12/3.74	   activate(n__0) -> 0
11.12/3.74	   activate(n__zeros) -> zeros
11.12/3.74	   activate(n__s(X)) -> s(X)
11.12/3.74	   activate(n__incr(X)) -> incr(X)
11.12/3.74	   activate(n__adx(X)) -> adx(X)
11.12/3.74	   activate(X) -> X
11.12/3.74	
11.12/3.74	S is empty.
11.12/3.74	Rewrite Strategy: FULL
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(9) InfiniteLowerBoundProof (FINISHED)
11.12/3.74	The following loop proves infinite runtime complexity:
11.12/3.74	
11.12/3.74	The rewrite sequence
11.12/3.74	
11.12/3.74	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))))))
11.12/3.74	
11.12/3.74	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
11.12/3.74	
11.12/3.74	The pumping substitution is [ ].
11.12/3.74	
11.12/3.74	The result substitution is [X / activate(X1_1), X1_1 / n__0].
11.12/3.74	
11.12/3.74	
11.12/3.74	
11.12/3.74	
11.12/3.74	----------------------------------------
11.12/3.74	
11.12/3.74	(10)
11.12/3.74	BOUNDS(INF, INF)
11.12/3.78	EOF