6.61/2.49	WORST_CASE(Omega(n^1), O(n^1))
6.61/2.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.61/2.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.61/2.50	
6.61/2.50	
6.61/2.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.61/2.50	
6.61/2.50	(0) CpxTRS
6.61/2.50	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
6.61/2.50	(2) CpxTRS
6.61/2.50	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
6.61/2.50	    (4) BOUNDS(1, n^1)
6.61/2.50	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
6.61/2.50	(6) TRS for Loop Detection
6.61/2.50	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
6.61/2.50	    (8) BEST
6.61/2.50	        (9) proven lower bound
6.61/2.50	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
6.61/2.50	            (11) BOUNDS(n^1, INF)
6.61/2.50	        (12) TRS for Loop Detection
6.61/2.50	
6.61/2.50	
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(0)
6.61/2.50	Obligation:
6.61/2.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.61/2.50	
6.61/2.50	
6.61/2.50	The TRS R consists of the following rules:
6.61/2.50	
6.61/2.50	   a(b(x)) -> b(b(a(x)))
6.61/2.50	
6.61/2.50	S is empty.
6.61/2.50	Rewrite Strategy: FULL
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(1) RelTrsToTrsProof (UPPER BOUND(ID))
6.61/2.50	transformed relative TRS to TRS
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(2)
6.61/2.50	Obligation:
6.61/2.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
6.61/2.50	
6.61/2.50	
6.61/2.50	The TRS R consists of the following rules:
6.61/2.50	
6.61/2.50	   a(b(x)) -> b(b(a(x)))
6.61/2.50	
6.61/2.50	S is empty.
6.61/2.50	Rewrite Strategy: FULL
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(3) CpxTrsMatchBoundsProof (FINISHED)
6.61/2.50	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
6.61/2.50	The certificate found is represented by the following graph.
6.61/2.50	
6.61/2.50	"[1, 2, 3, 4]
6.61/2.50	{(1,2,[a_1|0]), (1,3,[b_1|1]), (2,2,[b_1|0]), (3,4,[b_1|1]), (4,2,[a_1|1]), (4,3,[b_1|1])}"
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(4)
6.61/2.50	BOUNDS(1, n^1)
6.61/2.50	
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
6.61/2.50	Transformed a relative TRS into a decreasing-loop problem.
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(6)
6.61/2.50	Obligation:
6.61/2.50	Analyzing the following TRS for decreasing loops:
6.61/2.50	
6.61/2.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.61/2.50	
6.61/2.50	
6.61/2.50	The TRS R consists of the following rules:
6.61/2.50	
6.61/2.50	   a(b(x)) -> b(b(a(x)))
6.61/2.50	
6.61/2.50	S is empty.
6.61/2.50	Rewrite Strategy: FULL
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(7) DecreasingLoopProof (LOWER BOUND(ID))
6.61/2.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
6.61/2.50	
6.61/2.50	The rewrite sequence
6.61/2.50	
6.61/2.50	a(b(x)) ->^+ b(b(a(x)))
6.61/2.50	
6.61/2.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
6.61/2.50	
6.61/2.50	The pumping substitution is [x / b(x)].
6.61/2.50	
6.61/2.50	The result substitution is [ ].
6.61/2.50	
6.61/2.50	
6.61/2.50	
6.61/2.50	
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(8)
6.61/2.50	Complex Obligation (BEST)
6.61/2.50	
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(9)
6.61/2.50	Obligation:
6.61/2.50	Proved the lower bound n^1 for the following obligation:
6.61/2.50	
6.61/2.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.61/2.50	
6.61/2.50	
6.61/2.50	The TRS R consists of the following rules:
6.61/2.50	
6.61/2.50	   a(b(x)) -> b(b(a(x)))
6.61/2.50	
6.61/2.50	S is empty.
6.61/2.50	Rewrite Strategy: FULL
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(10) LowerBoundPropagationProof (FINISHED)
6.61/2.50	Propagated lower bound.
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(11)
6.61/2.50	BOUNDS(n^1, INF)
6.61/2.50	
6.61/2.50	----------------------------------------
6.61/2.50	
6.61/2.50	(12)
6.61/2.50	Obligation:
6.61/2.50	Analyzing the following TRS for decreasing loops:
6.61/2.50	
6.61/2.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.61/2.50	
6.61/2.50	
6.61/2.50	The TRS R consists of the following rules:
6.61/2.50	
6.61/2.50	   a(b(x)) -> b(b(a(x)))
6.61/2.50	
6.61/2.50	S is empty.
6.61/2.50	Rewrite Strategy: FULL
6.86/2.54	EOF