18.61/5.96	WORST_CASE(Omega(n^1), O(n^1))
18.61/5.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
18.61/5.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.61/5.97	
18.61/5.97	
18.61/5.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.61/5.97	
18.61/5.97	(0) CpxTRS
18.61/5.97	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
18.61/5.97	(2) CpxTRS
18.61/5.97	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
18.61/5.97	    (4) BOUNDS(1, n^1)
18.61/5.97	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
18.61/5.97	(6) TRS for Loop Detection
18.61/5.97	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
18.61/5.97	    (8) BEST
18.61/5.97	        (9) proven lower bound
18.61/5.97	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
18.61/5.97	            (11) BOUNDS(n^1, INF)
18.61/5.97	        (12) TRS for Loop Detection
18.61/5.97	
18.61/5.97	
18.61/5.97	----------------------------------------
18.61/5.97	
18.61/5.97	(0)
18.61/5.97	Obligation:
18.61/5.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.61/5.97	
18.61/5.97	
18.61/5.97	The TRS R consists of the following rules:
18.61/5.97	
18.61/5.97	   g(s(x)) -> f(x)
18.61/5.97	   f(0) -> s(0)
18.61/5.97	   f(s(x)) -> s(s(g(x)))
18.61/5.97	   g(0) -> 0
18.61/5.97	
18.61/5.97	S is empty.
18.61/5.97	Rewrite Strategy: FULL
18.61/5.97	----------------------------------------
18.61/5.97	
18.61/5.97	(1) RelTrsToTrsProof (UPPER BOUND(ID))
18.61/5.97	transformed relative TRS to TRS
18.61/5.97	----------------------------------------
18.61/5.97	
18.61/5.97	(2)
18.61/5.97	Obligation:
18.61/5.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
18.61/5.97	
18.61/5.97	
18.61/5.97	The TRS R consists of the following rules:
18.61/5.97	
18.61/5.97	   g(s(x)) -> f(x)
18.61/5.97	   f(0) -> s(0)
18.61/5.97	   f(s(x)) -> s(s(g(x)))
18.61/5.97	   g(0) -> 0
18.61/5.97	
18.61/5.97	S is empty.
18.61/5.97	Rewrite Strategy: FULL
18.61/5.97	----------------------------------------
18.61/5.97	
18.61/5.97	(3) CpxTrsMatchBoundsProof (FINISHED)
18.61/5.97	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
18.61/5.97	The certificate found is represented by the following graph.
18.61/5.97	
18.61/5.97	"[1, 2, 3, 4, 5]
18.61/5.97	{(1,2,[g_1|0, f_1|0, f_1|1, 0|1]), (1,3,[s_1|1]), (1,4,[s_1|1]), (2,2,[s_1|0, 0|0]), (3,2,[0|1]), (4,5,[s_1|1]), (5,2,[g_1|1, f_1|1, 0|1]), (5,3,[s_1|1]), (5,4,[s_1|1])}"
18.61/5.97	----------------------------------------
18.61/5.97	
18.61/5.97	(4)
18.61/5.97	BOUNDS(1, n^1)
18.61/5.97	
18.61/5.97	----------------------------------------
18.61/5.97	
18.61/5.97	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
18.61/5.97	Transformed a relative TRS into a decreasing-loop problem.
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(6)
18.79/5.97	Obligation:
18.79/5.97	Analyzing the following TRS for decreasing loops:
18.79/5.97	
18.79/5.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.79/5.97	
18.79/5.97	
18.79/5.97	The TRS R consists of the following rules:
18.79/5.97	
18.79/5.97	   g(s(x)) -> f(x)
18.79/5.97	   f(0) -> s(0)
18.79/5.97	   f(s(x)) -> s(s(g(x)))
18.79/5.97	   g(0) -> 0
18.79/5.97	
18.79/5.97	S is empty.
18.79/5.97	Rewrite Strategy: FULL
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(7) DecreasingLoopProof (LOWER BOUND(ID))
18.79/5.97	The following loop(s) give(s) rise to the lower bound Omega(n^1):
18.79/5.97	
18.79/5.97	The rewrite sequence
18.79/5.97	
18.79/5.97	g(s(s(x1_0))) ->^+ s(s(g(x1_0)))
18.79/5.97	
18.79/5.97	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
18.79/5.97	
18.79/5.97	The pumping substitution is [x1_0 / s(s(x1_0))].
18.79/5.97	
18.79/5.97	The result substitution is [ ].
18.79/5.97	
18.79/5.97	
18.79/5.97	
18.79/5.97	
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(8)
18.79/5.97	Complex Obligation (BEST)
18.79/5.97	
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(9)
18.79/5.97	Obligation:
18.79/5.97	Proved the lower bound n^1 for the following obligation:
18.79/5.97	
18.79/5.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.79/5.97	
18.79/5.97	
18.79/5.97	The TRS R consists of the following rules:
18.79/5.97	
18.79/5.97	   g(s(x)) -> f(x)
18.79/5.97	   f(0) -> s(0)
18.79/5.97	   f(s(x)) -> s(s(g(x)))
18.79/5.97	   g(0) -> 0
18.79/5.97	
18.79/5.97	S is empty.
18.79/5.97	Rewrite Strategy: FULL
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(10) LowerBoundPropagationProof (FINISHED)
18.79/5.97	Propagated lower bound.
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(11)
18.79/5.97	BOUNDS(n^1, INF)
18.79/5.97	
18.79/5.97	----------------------------------------
18.79/5.97	
18.79/5.97	(12)
18.79/5.97	Obligation:
18.79/5.97	Analyzing the following TRS for decreasing loops:
18.79/5.97	
18.79/5.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.79/5.97	
18.79/5.97	
18.79/5.97	The TRS R consists of the following rules:
18.79/5.97	
18.79/5.97	   g(s(x)) -> f(x)
18.79/5.97	   f(0) -> s(0)
18.79/5.97	   f(s(x)) -> s(s(g(x)))
18.79/5.97	   g(0) -> 0
18.79/5.97	
18.79/5.97	S is empty.
18.79/5.97	Rewrite Strategy: FULL
18.81/6.00	EOF