3.44/1.65	WORST_CASE(NON_POLY, ?)
3.44/1.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.44/1.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.65	
3.44/1.65	
3.44/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.65	
3.44/1.65	(0) CpxTRS
3.44/1.65	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.44/1.65	(2) TRS for Loop Detection
3.44/1.65	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.44/1.65	(4) BEST
3.44/1.65	    (5) proven lower bound
3.44/1.65	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.44/1.65	        (7) BOUNDS(n^1, INF)
3.44/1.65	    (8) TRS for Loop Detection
3.44/1.65	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.44/1.65	        (10) BOUNDS(INF, INF)
3.44/1.65	
3.44/1.65	
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(0)
3.44/1.65	Obligation:
3.44/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.65	
3.44/1.65	
3.44/1.65	The TRS R consists of the following rules:
3.44/1.65	
3.44/1.65	   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs)
3.44/1.65	   lookup(Nil, Cons(x, xs)) -> x
3.44/1.65	   run(e, p) -> intlookup(e, p)
3.44/1.65	   intlookup(e, p) -> intlookup(lookup(e, p), p)
3.44/1.65	
3.44/1.65	S is empty.
3.44/1.65	Rewrite Strategy: INNERMOST
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.44/1.65	Transformed a relative TRS into a decreasing-loop problem.
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(2)
3.44/1.65	Obligation:
3.44/1.65	Analyzing the following TRS for decreasing loops:
3.44/1.65	
3.44/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.65	
3.44/1.65	
3.44/1.65	The TRS R consists of the following rules:
3.44/1.65	
3.44/1.65	   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs)
3.44/1.65	   lookup(Nil, Cons(x, xs)) -> x
3.44/1.65	   run(e, p) -> intlookup(e, p)
3.44/1.65	   intlookup(e, p) -> intlookup(lookup(e, p), p)
3.44/1.65	
3.44/1.65	S is empty.
3.44/1.65	Rewrite Strategy: INNERMOST
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.44/1.65	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.44/1.65	
3.44/1.65	The rewrite sequence
3.44/1.65	
3.44/1.65	lookup(Cons(x', xs'), Cons(x, xs)) ->^+ lookup(xs', xs)
3.44/1.65	
3.44/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.44/1.65	
3.44/1.65	The pumping substitution is [xs' / Cons(x', xs'), xs / Cons(x, xs)].
3.44/1.65	
3.44/1.65	The result substitution is [ ].
3.44/1.65	
3.44/1.65	
3.44/1.65	
3.44/1.65	
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(4)
3.44/1.65	Complex Obligation (BEST)
3.44/1.65	
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(5)
3.44/1.65	Obligation:
3.44/1.65	Proved the lower bound n^1 for the following obligation:
3.44/1.65	
3.44/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.65	
3.44/1.65	
3.44/1.65	The TRS R consists of the following rules:
3.44/1.65	
3.44/1.65	   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs)
3.44/1.65	   lookup(Nil, Cons(x, xs)) -> x
3.44/1.65	   run(e, p) -> intlookup(e, p)
3.44/1.65	   intlookup(e, p) -> intlookup(lookup(e, p), p)
3.44/1.65	
3.44/1.65	S is empty.
3.44/1.65	Rewrite Strategy: INNERMOST
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(6) LowerBoundPropagationProof (FINISHED)
3.44/1.65	Propagated lower bound.
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(7)
3.44/1.65	BOUNDS(n^1, INF)
3.44/1.65	
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(8)
3.44/1.65	Obligation:
3.44/1.65	Analyzing the following TRS for decreasing loops:
3.44/1.65	
3.44/1.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.65	
3.44/1.65	
3.44/1.65	The TRS R consists of the following rules:
3.44/1.65	
3.44/1.65	   lookup(Cons(x', xs'), Cons(x, xs)) -> lookup(xs', xs)
3.44/1.65	   lookup(Nil, Cons(x, xs)) -> x
3.44/1.65	   run(e, p) -> intlookup(e, p)
3.44/1.65	   intlookup(e, p) -> intlookup(lookup(e, p), p)
3.44/1.65	
3.44/1.65	S is empty.
3.44/1.65	Rewrite Strategy: INNERMOST
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(9) InfiniteLowerBoundProof (FINISHED)
3.44/1.65	The following loop proves infinite runtime complexity:
3.44/1.65	
3.44/1.65	The rewrite sequence
3.44/1.65	
3.44/1.65	intlookup(e, p) ->^+ intlookup(lookup(e, p), p)
3.44/1.65	
3.44/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.44/1.65	
3.44/1.65	The pumping substitution is [ ].
3.44/1.65	
3.44/1.65	The result substitution is [e / lookup(e, p)].
3.44/1.65	
3.44/1.65	
3.44/1.65	
3.44/1.65	
3.44/1.65	----------------------------------------
3.44/1.65	
3.44/1.65	(10)
3.44/1.65	BOUNDS(INF, INF)
3.50/1.69	EOF