12.15/4.10	WORST_CASE(Omega(n^1), O(n^1))
12.15/4.11	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
12.15/4.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.15/4.11	
12.15/4.11	
12.15/4.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.15/4.11	
12.15/4.11	(0) CpxTRS
12.15/4.11	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
12.15/4.11	(2) CpxTRS
12.15/4.11	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
12.15/4.11	    (4) BOUNDS(1, n^1)
12.15/4.11	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
12.15/4.11	(6) TRS for Loop Detection
12.15/4.11	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
12.15/4.11	    (8) BEST
12.15/4.11	        (9) proven lower bound
12.15/4.11	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
12.15/4.11	            (11) BOUNDS(n^1, INF)
12.15/4.11	        (12) TRS for Loop Detection
12.15/4.11	
12.15/4.11	
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(0)
12.15/4.11	Obligation:
12.15/4.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.15/4.11	
12.15/4.11	
12.15/4.11	The TRS R consists of the following rules:
12.15/4.11	
12.15/4.11	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
12.15/4.11	   f(X) -> n__f(X)
12.15/4.11	   a -> n__a
12.15/4.11	   g(X) -> n__g(X)
12.15/4.11	   activate(n__f(X)) -> f(X)
12.15/4.11	   activate(n__a) -> a
12.15/4.11	   activate(n__g(X)) -> g(activate(X))
12.15/4.11	   activate(X) -> X
12.15/4.11	
12.15/4.11	S is empty.
12.15/4.11	Rewrite Strategy: INNERMOST
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(1) RelTrsToTrsProof (UPPER BOUND(ID))
12.15/4.11	transformed relative TRS to TRS
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(2)
12.15/4.11	Obligation:
12.15/4.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
12.15/4.11	
12.15/4.11	
12.15/4.11	The TRS R consists of the following rules:
12.15/4.11	
12.15/4.11	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
12.15/4.11	   f(X) -> n__f(X)
12.15/4.11	   a -> n__a
12.15/4.11	   g(X) -> n__g(X)
12.15/4.11	   activate(n__f(X)) -> f(X)
12.15/4.11	   activate(n__a) -> a
12.15/4.11	   activate(n__g(X)) -> g(activate(X))
12.15/4.11	   activate(X) -> X
12.15/4.11	
12.15/4.11	S is empty.
12.15/4.11	Rewrite Strategy: INNERMOST
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(3) CpxTrsMatchBoundsProof (FINISHED)
12.15/4.11	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
12.15/4.11	The certificate found is represented by the following graph.
12.15/4.11	
12.15/4.11	"[1, 2, 3, 4, 5, 6]
12.15/4.11	{(1,2,[f_1|0, a|0, g_1|0, activate_1|0, n__f_1|1, n__a|1, n__g_1|1, f_1|1, a|1, n__f_1|2, n__a|2]), (1,3,[f_1|1, n__f_1|2]), (1,6,[g_1|1, n__g_1|2]), (2,2,[n__f_1|0, n__a|0, n__g_1|0]), (3,4,[n__g_1|1]), (4,5,[n__f_1|1]), (5,2,[n__a|1]), (6,2,[activate_1|1, f_1|1, n__f_1|1, a|1, n__a|1, n__g_1|1, n__f_1|2, n__a|2]), (6,6,[g_1|1, n__g_1|2]), (6,3,[f_1|1, n__f_1|2])}"
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(4)
12.15/4.11	BOUNDS(1, n^1)
12.15/4.11	
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
12.15/4.11	Transformed a relative TRS into a decreasing-loop problem.
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(6)
12.15/4.11	Obligation:
12.15/4.11	Analyzing the following TRS for decreasing loops:
12.15/4.11	
12.15/4.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.15/4.11	
12.15/4.11	
12.15/4.11	The TRS R consists of the following rules:
12.15/4.11	
12.15/4.11	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
12.15/4.11	   f(X) -> n__f(X)
12.15/4.11	   a -> n__a
12.15/4.11	   g(X) -> n__g(X)
12.15/4.11	   activate(n__f(X)) -> f(X)
12.15/4.11	   activate(n__a) -> a
12.15/4.11	   activate(n__g(X)) -> g(activate(X))
12.15/4.11	   activate(X) -> X
12.15/4.11	
12.15/4.11	S is empty.
12.15/4.11	Rewrite Strategy: INNERMOST
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(7) DecreasingLoopProof (LOWER BOUND(ID))
12.15/4.11	The following loop(s) give(s) rise to the lower bound Omega(n^1):
12.15/4.11	
12.15/4.11	The rewrite sequence
12.15/4.11	
12.15/4.11	activate(n__g(X)) ->^+ g(activate(X))
12.15/4.11	
12.15/4.11	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
12.15/4.11	
12.15/4.11	The pumping substitution is [X / n__g(X)].
12.15/4.11	
12.15/4.11	The result substitution is [ ].
12.15/4.11	
12.15/4.11	
12.15/4.11	
12.15/4.11	
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(8)
12.15/4.11	Complex Obligation (BEST)
12.15/4.11	
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(9)
12.15/4.11	Obligation:
12.15/4.11	Proved the lower bound n^1 for the following obligation:
12.15/4.11	
12.15/4.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.15/4.11	
12.15/4.11	
12.15/4.11	The TRS R consists of the following rules:
12.15/4.11	
12.15/4.11	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
12.15/4.11	   f(X) -> n__f(X)
12.15/4.11	   a -> n__a
12.15/4.11	   g(X) -> n__g(X)
12.15/4.11	   activate(n__f(X)) -> f(X)
12.15/4.11	   activate(n__a) -> a
12.15/4.11	   activate(n__g(X)) -> g(activate(X))
12.15/4.11	   activate(X) -> X
12.15/4.11	
12.15/4.11	S is empty.
12.15/4.11	Rewrite Strategy: INNERMOST
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(10) LowerBoundPropagationProof (FINISHED)
12.15/4.11	Propagated lower bound.
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(11)
12.15/4.11	BOUNDS(n^1, INF)
12.15/4.11	
12.15/4.11	----------------------------------------
12.15/4.11	
12.15/4.11	(12)
12.15/4.11	Obligation:
12.15/4.11	Analyzing the following TRS for decreasing loops:
12.15/4.11	
12.15/4.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.15/4.11	
12.15/4.11	
12.15/4.11	The TRS R consists of the following rules:
12.15/4.11	
12.15/4.11	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
12.15/4.11	   f(X) -> n__f(X)
12.15/4.11	   a -> n__a
12.15/4.11	   g(X) -> n__g(X)
12.15/4.11	   activate(n__f(X)) -> f(X)
12.15/4.11	   activate(n__a) -> a
12.15/4.11	   activate(n__g(X)) -> g(activate(X))
12.15/4.11	   activate(X) -> X
12.15/4.11	
12.15/4.11	S is empty.
12.15/4.11	Rewrite Strategy: INNERMOST
12.43/4.16	EOF