7.05/2.58	WORST_CASE(Omega(n^1), O(n^1))
7.05/2.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.05/2.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.05/2.58	
7.05/2.58	
7.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.05/2.58	
7.05/2.58	(0) CpxTRS
7.05/2.58	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
7.05/2.58	(2) CpxTRS
7.05/2.58	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
7.05/2.58	    (4) BOUNDS(1, n^1)
7.05/2.58	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
7.05/2.58	(6) TRS for Loop Detection
7.05/2.58	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
7.05/2.58	    (8) BEST
7.05/2.58	        (9) proven lower bound
7.05/2.58	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
7.05/2.58	            (11) BOUNDS(n^1, INF)
7.05/2.58	        (12) TRS for Loop Detection
7.05/2.58	
7.05/2.58	
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(0)
7.05/2.58	Obligation:
7.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.05/2.58	
7.05/2.58	
7.05/2.58	The TRS R consists of the following rules:
7.05/2.58	
7.05/2.58	   f(g(x)) -> g(g(f(x)))
7.05/2.58	   f(g(x)) -> g(g(g(x)))
7.05/2.58	
7.05/2.58	S is empty.
7.05/2.58	Rewrite Strategy: INNERMOST
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(1) RelTrsToTrsProof (UPPER BOUND(ID))
7.05/2.58	transformed relative TRS to TRS
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(2)
7.05/2.58	Obligation:
7.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.05/2.58	
7.05/2.58	
7.05/2.58	The TRS R consists of the following rules:
7.05/2.58	
7.05/2.58	   f(g(x)) -> g(g(f(x)))
7.05/2.58	   f(g(x)) -> g(g(g(x)))
7.05/2.58	
7.05/2.58	S is empty.
7.05/2.58	Rewrite Strategy: INNERMOST
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(3) CpxTrsMatchBoundsProof (FINISHED)
7.05/2.58	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
7.05/2.58	The certificate found is represented by the following graph.
7.05/2.58	
7.05/2.58	"[1, 2, 3, 4, 5, 6]
7.05/2.58	{(1,2,[f_1|0]), (1,3,[g_1|1]), (1,5,[g_1|1]), (2,2,[g_1|0]), (3,4,[g_1|1]), (4,2,[f_1|1]), (4,3,[g_1|1]), (4,5,[g_1|1]), (5,6,[g_1|1]), (6,2,[g_1|1])}"
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(4)
7.05/2.58	BOUNDS(1, n^1)
7.05/2.58	
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
7.05/2.58	Transformed a relative TRS into a decreasing-loop problem.
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(6)
7.05/2.58	Obligation:
7.05/2.58	Analyzing the following TRS for decreasing loops:
7.05/2.58	
7.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.05/2.58	
7.05/2.58	
7.05/2.58	The TRS R consists of the following rules:
7.05/2.58	
7.05/2.58	   f(g(x)) -> g(g(f(x)))
7.05/2.58	   f(g(x)) -> g(g(g(x)))
7.05/2.58	
7.05/2.58	S is empty.
7.05/2.58	Rewrite Strategy: INNERMOST
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(7) DecreasingLoopProof (LOWER BOUND(ID))
7.05/2.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
7.05/2.58	
7.05/2.58	The rewrite sequence
7.05/2.58	
7.05/2.58	f(g(x)) ->^+ g(g(f(x)))
7.05/2.58	
7.05/2.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
7.05/2.58	
7.05/2.58	The pumping substitution is [x / g(x)].
7.05/2.58	
7.05/2.58	The result substitution is [ ].
7.05/2.58	
7.05/2.58	
7.05/2.58	
7.05/2.58	
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(8)
7.05/2.58	Complex Obligation (BEST)
7.05/2.58	
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(9)
7.05/2.58	Obligation:
7.05/2.58	Proved the lower bound n^1 for the following obligation:
7.05/2.58	
7.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.05/2.58	
7.05/2.58	
7.05/2.58	The TRS R consists of the following rules:
7.05/2.58	
7.05/2.58	   f(g(x)) -> g(g(f(x)))
7.05/2.58	   f(g(x)) -> g(g(g(x)))
7.05/2.58	
7.05/2.58	S is empty.
7.05/2.58	Rewrite Strategy: INNERMOST
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(10) LowerBoundPropagationProof (FINISHED)
7.05/2.58	Propagated lower bound.
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(11)
7.05/2.58	BOUNDS(n^1, INF)
7.05/2.58	
7.05/2.58	----------------------------------------
7.05/2.58	
7.05/2.58	(12)
7.05/2.58	Obligation:
7.05/2.58	Analyzing the following TRS for decreasing loops:
7.05/2.58	
7.05/2.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.05/2.58	
7.05/2.58	
7.05/2.58	The TRS R consists of the following rules:
7.05/2.58	
7.05/2.58	   f(g(x)) -> g(g(f(x)))
7.05/2.58	   f(g(x)) -> g(g(g(x)))
7.05/2.58	
7.05/2.58	S is empty.
7.05/2.58	Rewrite Strategy: INNERMOST
7.05/2.62	EOF