3.19/1.54	WORST_CASE(NON_POLY, ?)
3.19/1.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.19/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.19/1.55	
3.19/1.55	
3.19/1.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.19/1.55	
3.19/1.55	(0) CpxTRS
3.19/1.55	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.19/1.55	(2) TRS for Loop Detection
3.19/1.55	(3) DecreasingLoopProof [FINISHED, 0 ms]
3.19/1.55	(4) BOUNDS(EXP, INF)
3.19/1.55	
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(0)
3.19/1.55	Obligation:
3.19/1.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.19/1.55	
3.19/1.55	
3.19/1.55	The TRS R consists of the following rules:
3.19/1.55	
3.19/1.55	   qsort(nil) -> nil
3.19/1.55	   qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
3.19/1.55	   lowers(x, nil) -> nil
3.19/1.55	   lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
3.19/1.55	   greaters(x, nil) -> nil
3.19/1.55	   greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))
3.19/1.55	
3.19/1.55	S is empty.
3.19/1.55	Rewrite Strategy: INNERMOST
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.19/1.55	Transformed a relative TRS into a decreasing-loop problem.
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(2)
3.19/1.55	Obligation:
3.19/1.55	Analyzing the following TRS for decreasing loops:
3.19/1.55	
3.19/1.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.19/1.55	
3.19/1.55	
3.19/1.55	The TRS R consists of the following rules:
3.19/1.55	
3.19/1.55	   qsort(nil) -> nil
3.19/1.55	   qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
3.19/1.55	   lowers(x, nil) -> nil
3.19/1.55	   lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
3.19/1.55	   greaters(x, nil) -> nil
3.19/1.55	   greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))
3.19/1.55	
3.19/1.55	S is empty.
3.19/1.55	Rewrite Strategy: INNERMOST
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(3) DecreasingLoopProof (FINISHED)
3.19/1.55	The following loop(s) give(s) rise to the lower bound EXP:
3.19/1.55	
3.19/1.55	The rewrite sequence
3.19/1.55	
3.19/1.55	greaters(x, .(y, z)) ->^+ if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))
3.19/1.55	
3.19/1.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.19/1.55	
3.19/1.55	The pumping substitution is [z / .(y, z)].
3.19/1.55	
3.19/1.55	The result substitution is [ ].
3.19/1.55	
3.19/1.55	
3.19/1.55	
3.19/1.55	The rewrite sequence
3.19/1.55	
3.19/1.55	greaters(x, .(y, z)) ->^+ if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))
3.19/1.55	
3.19/1.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
3.19/1.55	
3.19/1.55	The pumping substitution is [z / .(y, z)].
3.19/1.55	
3.19/1.55	The result substitution is [ ].
3.19/1.55	
3.19/1.55	
3.19/1.55	
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(4)
3.19/1.55	BOUNDS(EXP, INF)
3.26/1.81	EOF