3.38/1.70	WORST_CASE(NON_POLY, ?)
3.68/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.68/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.68/1.73	
3.68/1.73	
3.68/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.68/1.73	
3.68/1.73	(0) CpxTRS
3.68/1.73	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.68/1.73	(2) TRS for Loop Detection
3.68/1.73	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.68/1.73	(4) BEST
3.68/1.73	    (5) proven lower bound
3.68/1.73	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.68/1.73	        (7) BOUNDS(n^1, INF)
3.68/1.73	    (8) TRS for Loop Detection
3.68/1.73	        (9) InfiniteLowerBoundProof [FINISHED, 10 ms]
3.68/1.73	        (10) BOUNDS(INF, INF)
3.68/1.73	
3.68/1.73	
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(0)
3.68/1.73	Obligation:
3.68/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.68/1.73	
3.68/1.73	
3.68/1.73	The TRS R consists of the following rules:
3.68/1.73	
3.68/1.73	   nats -> cons(0, n__incr(n__nats))
3.68/1.73	   pairs -> cons(0, n__incr(n__odds))
3.68/1.73	   odds -> incr(pairs)
3.68/1.73	   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
3.68/1.73	   head(cons(X, XS)) -> X
3.68/1.73	   tail(cons(X, XS)) -> activate(XS)
3.68/1.73	   incr(X) -> n__incr(X)
3.68/1.73	   nats -> n__nats
3.68/1.73	   odds -> n__odds
3.68/1.73	   activate(n__incr(X)) -> incr(activate(X))
3.68/1.73	   activate(n__nats) -> nats
3.68/1.73	   activate(n__odds) -> odds
3.68/1.73	   activate(X) -> X
3.68/1.73	
3.68/1.73	S is empty.
3.68/1.73	Rewrite Strategy: INNERMOST
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.68/1.73	Transformed a relative TRS into a decreasing-loop problem.
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(2)
3.68/1.73	Obligation:
3.68/1.73	Analyzing the following TRS for decreasing loops:
3.68/1.73	
3.68/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.68/1.73	
3.68/1.73	
3.68/1.73	The TRS R consists of the following rules:
3.68/1.73	
3.68/1.73	   nats -> cons(0, n__incr(n__nats))
3.68/1.73	   pairs -> cons(0, n__incr(n__odds))
3.68/1.73	   odds -> incr(pairs)
3.68/1.73	   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
3.68/1.73	   head(cons(X, XS)) -> X
3.68/1.73	   tail(cons(X, XS)) -> activate(XS)
3.68/1.73	   incr(X) -> n__incr(X)
3.68/1.73	   nats -> n__nats
3.68/1.73	   odds -> n__odds
3.68/1.73	   activate(n__incr(X)) -> incr(activate(X))
3.68/1.73	   activate(n__nats) -> nats
3.68/1.73	   activate(n__odds) -> odds
3.68/1.73	   activate(X) -> X
3.68/1.73	
3.68/1.73	S is empty.
3.68/1.73	Rewrite Strategy: INNERMOST
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.68/1.73	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.68/1.73	
3.68/1.73	The rewrite sequence
3.68/1.73	
3.68/1.73	activate(n__incr(X)) ->^+ incr(activate(X))
3.68/1.73	
3.68/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.68/1.73	
3.68/1.73	The pumping substitution is [X / n__incr(X)].
3.68/1.73	
3.68/1.73	The result substitution is [ ].
3.68/1.73	
3.68/1.73	
3.68/1.73	
3.68/1.73	
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(4)
3.68/1.73	Complex Obligation (BEST)
3.68/1.73	
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(5)
3.68/1.73	Obligation:
3.68/1.73	Proved the lower bound n^1 for the following obligation:
3.68/1.73	
3.68/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.68/1.73	
3.68/1.73	
3.68/1.73	The TRS R consists of the following rules:
3.68/1.73	
3.68/1.73	   nats -> cons(0, n__incr(n__nats))
3.68/1.73	   pairs -> cons(0, n__incr(n__odds))
3.68/1.73	   odds -> incr(pairs)
3.68/1.73	   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
3.68/1.73	   head(cons(X, XS)) -> X
3.68/1.73	   tail(cons(X, XS)) -> activate(XS)
3.68/1.73	   incr(X) -> n__incr(X)
3.68/1.73	   nats -> n__nats
3.68/1.73	   odds -> n__odds
3.68/1.73	   activate(n__incr(X)) -> incr(activate(X))
3.68/1.73	   activate(n__nats) -> nats
3.68/1.73	   activate(n__odds) -> odds
3.68/1.73	   activate(X) -> X
3.68/1.73	
3.68/1.73	S is empty.
3.68/1.73	Rewrite Strategy: INNERMOST
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(6) LowerBoundPropagationProof (FINISHED)
3.68/1.73	Propagated lower bound.
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(7)
3.68/1.73	BOUNDS(n^1, INF)
3.68/1.73	
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(8)
3.68/1.73	Obligation:
3.68/1.73	Analyzing the following TRS for decreasing loops:
3.68/1.73	
3.68/1.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.68/1.73	
3.68/1.73	
3.68/1.73	The TRS R consists of the following rules:
3.68/1.73	
3.68/1.73	   nats -> cons(0, n__incr(n__nats))
3.68/1.73	   pairs -> cons(0, n__incr(n__odds))
3.68/1.73	   odds -> incr(pairs)
3.68/1.73	   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
3.68/1.73	   head(cons(X, XS)) -> X
3.68/1.73	   tail(cons(X, XS)) -> activate(XS)
3.68/1.73	   incr(X) -> n__incr(X)
3.68/1.73	   nats -> n__nats
3.68/1.73	   odds -> n__odds
3.68/1.73	   activate(n__incr(X)) -> incr(activate(X))
3.68/1.73	   activate(n__nats) -> nats
3.68/1.73	   activate(n__odds) -> odds
3.68/1.73	   activate(X) -> X
3.68/1.73	
3.68/1.73	S is empty.
3.68/1.73	Rewrite Strategy: INNERMOST
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(9) InfiniteLowerBoundProof (FINISHED)
3.68/1.73	The following loop proves infinite runtime complexity:
3.68/1.73	
3.68/1.73	The rewrite sequence
3.68/1.73	
3.68/1.73	incr(cons(X, n__incr(n__nats))) ->^+ cons(s(X), n__incr(incr(cons(0, n__incr(n__nats)))))
3.68/1.73	
3.68/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
3.68/1.73	
3.68/1.73	The pumping substitution is [ ].
3.68/1.73	
3.68/1.73	The result substitution is [X / 0].
3.68/1.73	
3.68/1.73	
3.68/1.73	
3.68/1.73	
3.68/1.73	----------------------------------------
3.68/1.73	
3.68/1.73	(10)
3.68/1.73	BOUNDS(INF, INF)
3.68/1.76	EOF