6.67/2.55	WORST_CASE(Omega(n^1), O(n^1))
8.59/3.06	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
8.59/3.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.59/3.06	
8.59/3.06	
8.59/3.06	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.06	
8.59/3.06	(0) CpxTRS
8.59/3.06	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
8.59/3.06	(2) CpxTRS
8.59/3.06	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
8.59/3.06	    (4) BOUNDS(1, n^1)
8.59/3.06	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
8.59/3.06	(6) TRS for Loop Detection
8.59/3.06	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
8.59/3.06	    (8) BEST
8.59/3.06	        (9) proven lower bound
8.59/3.06	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
8.59/3.06	            (11) BOUNDS(n^1, INF)
8.59/3.06	        (12) TRS for Loop Detection
8.59/3.06	
8.59/3.06	
8.59/3.06	----------------------------------------
8.59/3.06	
8.59/3.06	(0)
8.59/3.06	Obligation:
8.59/3.06	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.06	
8.59/3.06	
8.59/3.06	The TRS R consists of the following rules:
8.59/3.06	
8.59/3.06	   f(g(X)) -> f(X)
8.59/3.06	
8.59/3.06	S is empty.
8.59/3.06	Rewrite Strategy: INNERMOST
8.59/3.06	----------------------------------------
8.59/3.06	
8.59/3.06	(1) RelTrsToTrsProof (UPPER BOUND(ID))
8.59/3.07	transformed relative TRS to TRS
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(2)
8.59/3.07	Obligation:
8.59/3.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
8.59/3.07	
8.59/3.07	
8.59/3.07	The TRS R consists of the following rules:
8.59/3.07	
8.59/3.07	   f(g(X)) -> f(X)
8.59/3.07	
8.59/3.07	S is empty.
8.59/3.07	Rewrite Strategy: INNERMOST
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(3) CpxTrsMatchBoundsProof (FINISHED)
8.59/3.07	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
8.59/3.07	The certificate found is represented by the following graph.
8.59/3.07	
8.59/3.07	"[1, 2]
8.59/3.07	{(1,2,[f_1|0, f_1|1]), (2,2,[g_1|0])}"
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(4)
8.59/3.07	BOUNDS(1, n^1)
8.59/3.07	
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
8.59/3.07	Transformed a relative TRS into a decreasing-loop problem.
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(6)
8.59/3.07	Obligation:
8.59/3.07	Analyzing the following TRS for decreasing loops:
8.59/3.07	
8.59/3.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.07	
8.59/3.07	
8.59/3.07	The TRS R consists of the following rules:
8.59/3.07	
8.59/3.07	   f(g(X)) -> f(X)
8.59/3.07	
8.59/3.07	S is empty.
8.59/3.07	Rewrite Strategy: INNERMOST
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(7) DecreasingLoopProof (LOWER BOUND(ID))
8.59/3.07	The following loop(s) give(s) rise to the lower bound Omega(n^1):
8.59/3.07	
8.59/3.07	The rewrite sequence
8.59/3.07	
8.59/3.07	f(g(X)) ->^+ f(X)
8.59/3.07	
8.59/3.07	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
8.59/3.07	
8.59/3.07	The pumping substitution is [X / g(X)].
8.59/3.07	
8.59/3.07	The result substitution is [ ].
8.59/3.07	
8.59/3.07	
8.59/3.07	
8.59/3.07	
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(8)
8.59/3.07	Complex Obligation (BEST)
8.59/3.07	
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(9)
8.59/3.07	Obligation:
8.59/3.07	Proved the lower bound n^1 for the following obligation:
8.59/3.07	
8.59/3.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.07	
8.59/3.07	
8.59/3.07	The TRS R consists of the following rules:
8.59/3.07	
8.59/3.07	   f(g(X)) -> f(X)
8.59/3.07	
8.59/3.07	S is empty.
8.59/3.07	Rewrite Strategy: INNERMOST
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(10) LowerBoundPropagationProof (FINISHED)
8.59/3.07	Propagated lower bound.
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(11)
8.59/3.07	BOUNDS(n^1, INF)
8.59/3.07	
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(12)
8.59/3.07	Obligation:
8.59/3.07	Analyzing the following TRS for decreasing loops:
8.59/3.07	
8.59/3.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.07	
8.59/3.07	
8.59/3.07	The TRS R consists of the following rules:
8.59/3.07	
8.59/3.07	   f(g(X)) -> f(X)
8.59/3.07	
8.59/3.07	S is empty.
8.59/3.07	Rewrite Strategy: INNERMOST
8.59/3.10	EOF