2.93/1.56	WORST_CASE(NON_POLY, ?)
3.21/1.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.21/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.21/1.57	
3.21/1.57	
3.21/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.21/1.57	
3.21/1.57	(0) CpxTRS
3.21/1.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.21/1.57	(2) TRS for Loop Detection
3.21/1.57	(3) DecreasingLoopProof [FINISHED, 0 ms]
3.21/1.57	(4) BOUNDS(EXP, INF)
3.21/1.57	
3.21/1.57	
3.21/1.57	----------------------------------------
3.21/1.57	
3.21/1.57	(0)
3.21/1.57	Obligation:
3.21/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.21/1.57	
3.21/1.57	
3.21/1.57	The TRS R consists of the following rules:
3.21/1.57	
3.21/1.57	   bsort(nil) -> nil
3.21/1.57	   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
3.21/1.57	   bubble(nil) -> nil
3.21/1.57	   bubble(.(x, nil)) -> .(x, nil)
3.21/1.57	   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
3.21/1.57	   last(nil) -> 0
3.21/1.57	   last(.(x, nil)) -> x
3.21/1.57	   last(.(x, .(y, z))) -> last(.(y, z))
3.21/1.57	   butlast(nil) -> nil
3.21/1.57	   butlast(.(x, nil)) -> nil
3.21/1.57	   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))
3.21/1.57	
3.21/1.57	S is empty.
3.21/1.57	Rewrite Strategy: INNERMOST
3.21/1.57	----------------------------------------
3.21/1.57	
3.21/1.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.21/1.57	Transformed a relative TRS into a decreasing-loop problem.
3.21/1.57	----------------------------------------
3.21/1.57	
3.21/1.57	(2)
3.21/1.57	Obligation:
3.21/1.57	Analyzing the following TRS for decreasing loops:
3.21/1.57	
3.21/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.21/1.57	
3.21/1.57	
3.21/1.57	The TRS R consists of the following rules:
3.21/1.57	
3.21/1.57	   bsort(nil) -> nil
3.21/1.57	   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
3.21/1.57	   bubble(nil) -> nil
3.21/1.57	   bubble(.(x, nil)) -> .(x, nil)
3.21/1.57	   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
3.21/1.57	   last(nil) -> 0
3.21/1.57	   last(.(x, nil)) -> x
3.21/1.57	   last(.(x, .(y, z))) -> last(.(y, z))
3.21/1.57	   butlast(nil) -> nil
3.21/1.57	   butlast(.(x, nil)) -> nil
3.21/1.57	   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))
3.21/1.57	
3.21/1.57	S is empty.
3.21/1.57	Rewrite Strategy: INNERMOST
3.21/1.57	----------------------------------------
3.21/1.57	
3.21/1.57	(3) DecreasingLoopProof (FINISHED)
3.21/1.57	The following loop(s) give(s) rise to the lower bound EXP:
3.21/1.57	
3.21/1.57	The rewrite sequence
3.21/1.57	
3.21/1.57	bubble(.(x, .(y, z))) ->^+ if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
3.21/1.57	
3.21/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
3.21/1.57	
3.21/1.57	The pumping substitution is [z / .(y, z)].
3.21/1.57	
3.21/1.57	The result substitution is [ ].
3.21/1.57	
3.21/1.57	
3.21/1.57	
3.21/1.57	The rewrite sequence
3.21/1.57	
3.21/1.57	bubble(.(x, .(y, z))) ->^+ if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
3.21/1.57	
3.21/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
3.21/1.57	
3.21/1.57	The pumping substitution is [z / .(y, z)].
3.21/1.57	
3.21/1.57	The result substitution is [x / y].
3.21/1.57	
3.21/1.57	
3.21/1.57	
3.21/1.57	
3.21/1.57	----------------------------------------
3.21/1.57	
3.21/1.57	(4)
3.21/1.57	BOUNDS(EXP, INF)
3.23/1.60	EOF