18.35/6.40	WORST_CASE(Omega(n^1), O(n^1))
18.35/6.41	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
18.35/6.41	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.35/6.41	
18.35/6.41	
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.35/6.41	
18.35/6.41	(0) CpxTRS
18.35/6.41	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
18.35/6.41	(2) CpxTRS
18.35/6.41	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
18.35/6.41	    (4) CpxTRS
18.35/6.41	    (5) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
18.35/6.41	    (6) BOUNDS(1, n^1)
18.35/6.41	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
18.35/6.41	(8) TRS for Loop Detection
18.35/6.41	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
18.35/6.41	    (10) BEST
18.35/6.41	        (11) proven lower bound
18.35/6.41	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
18.35/6.41	            (13) BOUNDS(n^1, INF)
18.35/6.41	        (14) TRS for Loop Detection
18.35/6.41	
18.35/6.41	
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(0)
18.35/6.41	Obligation:
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.35/6.41	
18.35/6.41	
18.35/6.41	The TRS R consists of the following rules:
18.35/6.41	
18.35/6.41	   f(s(0), g(x)) -> f(x, g(x))
18.35/6.41	   g(s(x)) -> g(x)
18.35/6.41	
18.35/6.41	S is empty.
18.35/6.41	Rewrite Strategy: FULL
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(1) DependencyGraphProof (UPPER BOUND(ID))
18.35/6.41	The following rules are not reachable from basic terms in the dependency graph and can be removed:
18.35/6.41	
18.35/6.41	f(s(0), g(x)) -> f(x, g(x))
18.35/6.41	
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(2)
18.35/6.41	Obligation:
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
18.35/6.41	
18.35/6.41	
18.35/6.41	The TRS R consists of the following rules:
18.35/6.41	
18.35/6.41	   g(s(x)) -> g(x)
18.35/6.41	
18.35/6.41	S is empty.
18.35/6.41	Rewrite Strategy: FULL
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(3) RelTrsToTrsProof (UPPER BOUND(ID))
18.35/6.41	transformed relative TRS to TRS
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(4)
18.35/6.41	Obligation:
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
18.35/6.41	
18.35/6.41	
18.35/6.41	The TRS R consists of the following rules:
18.35/6.41	
18.35/6.41	   g(s(x)) -> g(x)
18.35/6.41	
18.35/6.41	S is empty.
18.35/6.41	Rewrite Strategy: FULL
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(5) CpxTrsMatchBoundsProof (FINISHED)
18.35/6.41	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
18.35/6.41	The certificate found is represented by the following graph.
18.35/6.41	
18.35/6.41	"[1, 2]
18.35/6.41	{(1,2,[g_1|0, g_1|1]), (2,2,[s_1|0])}"
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(6)
18.35/6.41	BOUNDS(1, n^1)
18.35/6.41	
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
18.35/6.41	Transformed a relative TRS into a decreasing-loop problem.
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(8)
18.35/6.41	Obligation:
18.35/6.41	Analyzing the following TRS for decreasing loops:
18.35/6.41	
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.35/6.41	
18.35/6.41	
18.35/6.41	The TRS R consists of the following rules:
18.35/6.41	
18.35/6.41	   f(s(0), g(x)) -> f(x, g(x))
18.35/6.41	   g(s(x)) -> g(x)
18.35/6.41	
18.35/6.41	S is empty.
18.35/6.41	Rewrite Strategy: FULL
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(9) DecreasingLoopProof (LOWER BOUND(ID))
18.35/6.41	The following loop(s) give(s) rise to the lower bound Omega(n^1):
18.35/6.41	
18.35/6.41	The rewrite sequence
18.35/6.41	
18.35/6.41	g(s(x)) ->^+ g(x)
18.35/6.41	
18.35/6.41	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
18.35/6.41	
18.35/6.41	The pumping substitution is [x / s(x)].
18.35/6.41	
18.35/6.41	The result substitution is [ ].
18.35/6.41	
18.35/6.41	
18.35/6.41	
18.35/6.41	
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(10)
18.35/6.41	Complex Obligation (BEST)
18.35/6.41	
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(11)
18.35/6.41	Obligation:
18.35/6.41	Proved the lower bound n^1 for the following obligation:
18.35/6.41	
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.35/6.41	
18.35/6.41	
18.35/6.41	The TRS R consists of the following rules:
18.35/6.41	
18.35/6.41	   f(s(0), g(x)) -> f(x, g(x))
18.35/6.41	   g(s(x)) -> g(x)
18.35/6.41	
18.35/6.41	S is empty.
18.35/6.41	Rewrite Strategy: FULL
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(12) LowerBoundPropagationProof (FINISHED)
18.35/6.41	Propagated lower bound.
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(13)
18.35/6.41	BOUNDS(n^1, INF)
18.35/6.41	
18.35/6.41	----------------------------------------
18.35/6.41	
18.35/6.41	(14)
18.35/6.41	Obligation:
18.35/6.41	Analyzing the following TRS for decreasing loops:
18.35/6.41	
18.35/6.41	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.35/6.41	
18.35/6.41	
18.35/6.41	The TRS R consists of the following rules:
18.35/6.41	
18.35/6.41	   f(s(0), g(x)) -> f(x, g(x))
18.35/6.41	   g(s(x)) -> g(x)
18.35/6.41	
18.35/6.41	S is empty.
18.35/6.41	Rewrite Strategy: FULL
18.53/6.44	EOF