6.40/2.48	WORST_CASE(Omega(n^1), O(n^1))
6.81/2.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.81/2.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.81/2.49	
6.81/2.49	
6.81/2.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.81/2.49	
6.81/2.49	(0) CpxTRS
6.81/2.49	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
6.81/2.49	(2) CpxTRS
6.81/2.49	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
6.81/2.49	    (4) BOUNDS(1, n^1)
6.81/2.49	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
6.81/2.49	(6) TRS for Loop Detection
6.81/2.49	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
6.81/2.49	    (8) BEST
6.81/2.49	        (9) proven lower bound
6.81/2.49	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
6.81/2.49	            (11) BOUNDS(n^1, INF)
6.81/2.49	        (12) TRS for Loop Detection
6.81/2.49	
6.81/2.49	
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(0)
6.81/2.49	Obligation:
6.81/2.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.81/2.49	
6.81/2.49	
6.81/2.49	The TRS R consists of the following rules:
6.81/2.49	
6.81/2.49	   f(g(X)) -> f(X)
6.81/2.49	
6.81/2.49	S is empty.
6.81/2.49	Rewrite Strategy: FULL
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(1) RelTrsToTrsProof (UPPER BOUND(ID))
6.81/2.49	transformed relative TRS to TRS
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(2)
6.81/2.49	Obligation:
6.81/2.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
6.81/2.49	
6.81/2.49	
6.81/2.49	The TRS R consists of the following rules:
6.81/2.49	
6.81/2.49	   f(g(X)) -> f(X)
6.81/2.49	
6.81/2.49	S is empty.
6.81/2.49	Rewrite Strategy: FULL
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(3) CpxTrsMatchBoundsProof (FINISHED)
6.81/2.49	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
6.81/2.49	The certificate found is represented by the following graph.
6.81/2.49	
6.81/2.49	"[1, 2]
6.81/2.49	{(1,2,[f_1|0, f_1|1]), (2,2,[g_1|0])}"
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(4)
6.81/2.49	BOUNDS(1, n^1)
6.81/2.49	
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
6.81/2.49	Transformed a relative TRS into a decreasing-loop problem.
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(6)
6.81/2.49	Obligation:
6.81/2.49	Analyzing the following TRS for decreasing loops:
6.81/2.49	
6.81/2.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.81/2.49	
6.81/2.49	
6.81/2.49	The TRS R consists of the following rules:
6.81/2.49	
6.81/2.49	   f(g(X)) -> f(X)
6.81/2.49	
6.81/2.49	S is empty.
6.81/2.49	Rewrite Strategy: FULL
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(7) DecreasingLoopProof (LOWER BOUND(ID))
6.81/2.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
6.81/2.49	
6.81/2.49	The rewrite sequence
6.81/2.49	
6.81/2.49	f(g(X)) ->^+ f(X)
6.81/2.49	
6.81/2.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
6.81/2.49	
6.81/2.49	The pumping substitution is [X / g(X)].
6.81/2.49	
6.81/2.49	The result substitution is [ ].
6.81/2.49	
6.81/2.49	
6.81/2.49	
6.81/2.49	
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(8)
6.81/2.49	Complex Obligation (BEST)
6.81/2.49	
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(9)
6.81/2.49	Obligation:
6.81/2.49	Proved the lower bound n^1 for the following obligation:
6.81/2.49	
6.81/2.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.81/2.49	
6.81/2.49	
6.81/2.49	The TRS R consists of the following rules:
6.81/2.49	
6.81/2.49	   f(g(X)) -> f(X)
6.81/2.49	
6.81/2.49	S is empty.
6.81/2.49	Rewrite Strategy: FULL
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(10) LowerBoundPropagationProof (FINISHED)
6.81/2.49	Propagated lower bound.
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(11)
6.81/2.49	BOUNDS(n^1, INF)
6.81/2.49	
6.81/2.49	----------------------------------------
6.81/2.49	
6.81/2.49	(12)
6.81/2.49	Obligation:
6.81/2.49	Analyzing the following TRS for decreasing loops:
6.81/2.49	
6.81/2.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.81/2.49	
6.81/2.49	
6.81/2.49	The TRS R consists of the following rules:
6.81/2.49	
6.81/2.49	   f(g(X)) -> f(X)
6.81/2.49	
6.81/2.49	S is empty.
6.81/2.49	Rewrite Strategy: FULL
6.84/2.53	EOF