2.96/1.61	WORST_CASE(NON_POLY, ?)
2.96/1.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.96/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.96/1.61	
2.96/1.61	
2.96/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.96/1.61	
2.96/1.61	(0) CpxTRS
2.96/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.96/1.61	(2) TRS for Loop Detection
2.96/1.61	(3) DecreasingLoopProof [FINISHED, 0 ms]
2.96/1.61	(4) BOUNDS(EXP, INF)
2.96/1.61	
2.96/1.61	
2.96/1.61	----------------------------------------
2.96/1.61	
2.96/1.61	(0)
2.96/1.61	Obligation:
2.96/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.96/1.61	
2.96/1.61	
2.96/1.61	The TRS R consists of the following rules:
2.96/1.61	
2.96/1.61	   bsort(nil) -> nil
2.96/1.61	   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
2.96/1.61	   bubble(nil) -> nil
2.96/1.61	   bubble(.(x, nil)) -> .(x, nil)
2.96/1.61	   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
2.96/1.61	   last(nil) -> 0
2.96/1.61	   last(.(x, nil)) -> x
2.96/1.61	   last(.(x, .(y, z))) -> last(.(y, z))
2.96/1.61	   butlast(nil) -> nil
2.96/1.61	   butlast(.(x, nil)) -> nil
2.96/1.61	   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))
2.96/1.61	
2.96/1.61	S is empty.
2.96/1.61	Rewrite Strategy: FULL
2.96/1.61	----------------------------------------
2.96/1.61	
2.96/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.96/1.61	Transformed a relative TRS into a decreasing-loop problem.
2.96/1.61	----------------------------------------
2.96/1.61	
2.96/1.61	(2)
2.96/1.61	Obligation:
2.96/1.61	Analyzing the following TRS for decreasing loops:
2.96/1.61	
2.96/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
2.96/1.61	
2.96/1.61	
2.96/1.61	The TRS R consists of the following rules:
2.96/1.61	
2.96/1.61	   bsort(nil) -> nil
2.96/1.61	   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
2.96/1.61	   bubble(nil) -> nil
2.96/1.61	   bubble(.(x, nil)) -> .(x, nil)
2.96/1.61	   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
2.96/1.61	   last(nil) -> 0
2.96/1.61	   last(.(x, nil)) -> x
2.96/1.61	   last(.(x, .(y, z))) -> last(.(y, z))
2.96/1.61	   butlast(nil) -> nil
2.96/1.61	   butlast(.(x, nil)) -> nil
2.96/1.61	   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))
2.96/1.61	
2.96/1.61	S is empty.
2.96/1.61	Rewrite Strategy: FULL
2.96/1.61	----------------------------------------
2.96/1.61	
2.96/1.61	(3) DecreasingLoopProof (FINISHED)
2.96/1.61	The following loop(s) give(s) rise to the lower bound EXP:
2.96/1.61	
2.96/1.61	The rewrite sequence
2.96/1.61	
2.96/1.61	bubble(.(x, .(y, z))) ->^+ if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
2.96/1.61	
2.96/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
2.96/1.61	
2.96/1.61	The pumping substitution is [z / .(y, z)].
2.96/1.61	
2.96/1.61	The result substitution is [ ].
2.96/1.61	
2.96/1.61	
2.96/1.61	
2.96/1.61	The rewrite sequence
2.96/1.61	
2.96/1.61	bubble(.(x, .(y, z))) ->^+ if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
2.96/1.61	
2.96/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
2.96/1.61	
2.96/1.61	The pumping substitution is [z / .(y, z)].
2.96/1.61	
2.96/1.61	The result substitution is [x / y].
2.96/1.61	
2.96/1.61	
2.96/1.61	
2.96/1.61	
2.96/1.61	----------------------------------------
2.96/1.61	
2.96/1.61	(4)
2.96/1.61	BOUNDS(EXP, INF)
3.26/1.65	EOF