7.17/2.63	WORST_CASE(Omega(n^1), O(n^1))
7.17/2.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.17/2.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.17/2.63	
7.17/2.63	
7.17/2.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.17/2.63	
7.17/2.63	(0) CpxTRS
7.17/2.63	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
7.17/2.63	(2) CpxTRS
7.17/2.63	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
7.17/2.63	    (4) BOUNDS(1, n^1)
7.17/2.63	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
7.17/2.63	(6) TRS for Loop Detection
7.17/2.63	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
7.17/2.63	    (8) BEST
7.17/2.63	        (9) proven lower bound
7.17/2.63	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
7.17/2.63	            (11) BOUNDS(n^1, INF)
7.17/2.63	        (12) TRS for Loop Detection
7.17/2.63	
7.17/2.63	
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(0)
7.17/2.63	Obligation:
7.17/2.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.17/2.63	
7.17/2.63	
7.17/2.63	The TRS R consists of the following rules:
7.17/2.63	
7.17/2.63	   a(b(x)) -> b(a(x))
7.17/2.63	   a(c(x)) -> x
7.17/2.63	
7.17/2.63	S is empty.
7.17/2.63	Rewrite Strategy: FULL
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(1) RelTrsToTrsProof (UPPER BOUND(ID))
7.17/2.63	transformed relative TRS to TRS
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(2)
7.17/2.63	Obligation:
7.17/2.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.17/2.63	
7.17/2.63	
7.17/2.63	The TRS R consists of the following rules:
7.17/2.63	
7.17/2.63	   a(b(x)) -> b(a(x))
7.17/2.63	   a(c(x)) -> x
7.17/2.63	
7.17/2.63	S is empty.
7.17/2.63	Rewrite Strategy: FULL
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(3) CpxTrsMatchBoundsProof (FINISHED)
7.17/2.63	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
7.17/2.63	The certificate found is represented by the following graph.
7.17/2.63	
7.17/2.63	"[1, 2, 3]
7.17/2.63	{(1,2,[a_1|0, b_1|1, c_1|1]), (1,3,[b_1|1]), (2,2,[b_1|0, c_1|0]), (3,2,[a_1|1, b_1|1, c_1|1]), (3,3,[b_1|1])}"
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(4)
7.17/2.63	BOUNDS(1, n^1)
7.17/2.63	
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
7.17/2.63	Transformed a relative TRS into a decreasing-loop problem.
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(6)
7.17/2.63	Obligation:
7.17/2.63	Analyzing the following TRS for decreasing loops:
7.17/2.63	
7.17/2.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.17/2.63	
7.17/2.63	
7.17/2.63	The TRS R consists of the following rules:
7.17/2.63	
7.17/2.63	   a(b(x)) -> b(a(x))
7.17/2.63	   a(c(x)) -> x
7.17/2.63	
7.17/2.63	S is empty.
7.17/2.63	Rewrite Strategy: FULL
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(7) DecreasingLoopProof (LOWER BOUND(ID))
7.17/2.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
7.17/2.63	
7.17/2.63	The rewrite sequence
7.17/2.63	
7.17/2.63	a(b(x)) ->^+ b(a(x))
7.17/2.63	
7.17/2.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
7.17/2.63	
7.17/2.63	The pumping substitution is [x / b(x)].
7.17/2.63	
7.17/2.63	The result substitution is [ ].
7.17/2.63	
7.17/2.63	
7.17/2.63	
7.17/2.63	
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(8)
7.17/2.63	Complex Obligation (BEST)
7.17/2.63	
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(9)
7.17/2.63	Obligation:
7.17/2.63	Proved the lower bound n^1 for the following obligation:
7.17/2.63	
7.17/2.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.17/2.63	
7.17/2.63	
7.17/2.63	The TRS R consists of the following rules:
7.17/2.63	
7.17/2.63	   a(b(x)) -> b(a(x))
7.17/2.63	   a(c(x)) -> x
7.17/2.63	
7.17/2.63	S is empty.
7.17/2.63	Rewrite Strategy: FULL
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(10) LowerBoundPropagationProof (FINISHED)
7.17/2.63	Propagated lower bound.
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(11)
7.17/2.63	BOUNDS(n^1, INF)
7.17/2.63	
7.17/2.63	----------------------------------------
7.17/2.63	
7.17/2.63	(12)
7.17/2.63	Obligation:
7.17/2.63	Analyzing the following TRS for decreasing loops:
7.17/2.63	
7.17/2.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.17/2.63	
7.17/2.63	
7.17/2.63	The TRS R consists of the following rules:
7.17/2.63	
7.17/2.63	   a(b(x)) -> b(a(x))
7.17/2.63	   a(c(x)) -> x
7.17/2.63	
7.17/2.63	S is empty.
7.17/2.63	Rewrite Strategy: FULL
7.44/2.74	EOF