19.19/6.66	WORST_CASE(Omega(n^1), O(n^1))
19.19/6.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.19/6.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.19/6.67	
19.19/6.67	
19.19/6.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.19/6.67	
19.19/6.67	(0) CpxTRS
19.19/6.67	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
19.19/6.67	(2) CpxTRS
19.19/6.67	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
19.19/6.67	    (4) BOUNDS(1, n^1)
19.19/6.67	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
19.19/6.67	(6) TRS for Loop Detection
19.19/6.67	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
19.19/6.67	    (8) BEST
19.19/6.67	        (9) proven lower bound
19.19/6.67	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
19.19/6.67	            (11) BOUNDS(n^1, INF)
19.19/6.67	        (12) TRS for Loop Detection
19.19/6.67	
19.19/6.67	
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(0)
19.19/6.67	Obligation:
19.19/6.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.19/6.67	
19.19/6.67	
19.19/6.67	The TRS R consists of the following rules:
19.19/6.67	
19.19/6.67	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
19.19/6.67	   f(X) -> n__f(X)
19.19/6.67	   a -> n__a
19.19/6.67	   g(X) -> n__g(X)
19.19/6.67	   activate(n__f(X)) -> f(X)
19.19/6.67	   activate(n__a) -> a
19.19/6.67	   activate(n__g(X)) -> g(activate(X))
19.19/6.67	   activate(X) -> X
19.19/6.67	
19.19/6.67	S is empty.
19.19/6.67	Rewrite Strategy: FULL
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(1) RelTrsToTrsProof (UPPER BOUND(ID))
19.19/6.67	transformed relative TRS to TRS
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(2)
19.19/6.67	Obligation:
19.19/6.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.19/6.67	
19.19/6.67	
19.19/6.67	The TRS R consists of the following rules:
19.19/6.67	
19.19/6.67	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
19.19/6.67	   f(X) -> n__f(X)
19.19/6.67	   a -> n__a
19.19/6.67	   g(X) -> n__g(X)
19.19/6.67	   activate(n__f(X)) -> f(X)
19.19/6.67	   activate(n__a) -> a
19.19/6.67	   activate(n__g(X)) -> g(activate(X))
19.19/6.67	   activate(X) -> X
19.19/6.67	
19.19/6.67	S is empty.
19.19/6.67	Rewrite Strategy: FULL
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(3) CpxTrsMatchBoundsProof (FINISHED)
19.19/6.67	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
19.19/6.67	The certificate found is represented by the following graph.
19.19/6.67	
19.19/6.67	"[1, 2, 7, 8, 9, 10]
19.19/6.67	{(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]), (1,7,[f_1|1, n__f_1|2]), (1,10,[g_1|1, n__g_1|2]), (2,2,[n__f_1|0, n__a|0, n__g_1|0]), (7,8,[n__g_1|1]), (8,9,[n__f_1|1]), (9,2,[n__a|1]), (10,2,[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]), (10,10,[g_1|1, n__g_1|2]), (10,7,[f_1|1, n__f_1|2])}"
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(4)
19.19/6.67	BOUNDS(1, n^1)
19.19/6.67	
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
19.19/6.67	Transformed a relative TRS into a decreasing-loop problem.
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(6)
19.19/6.67	Obligation:
19.19/6.67	Analyzing the following TRS for decreasing loops:
19.19/6.67	
19.19/6.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.19/6.67	
19.19/6.67	
19.19/6.67	The TRS R consists of the following rules:
19.19/6.67	
19.19/6.67	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
19.19/6.67	   f(X) -> n__f(X)
19.19/6.67	   a -> n__a
19.19/6.67	   g(X) -> n__g(X)
19.19/6.67	   activate(n__f(X)) -> f(X)
19.19/6.67	   activate(n__a) -> a
19.19/6.67	   activate(n__g(X)) -> g(activate(X))
19.19/6.67	   activate(X) -> X
19.19/6.67	
19.19/6.67	S is empty.
19.19/6.67	Rewrite Strategy: FULL
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(7) DecreasingLoopProof (LOWER BOUND(ID))
19.19/6.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
19.19/6.67	
19.19/6.67	The rewrite sequence
19.19/6.67	
19.19/6.67	activate(n__g(X)) ->^+ g(activate(X))
19.19/6.67	
19.19/6.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
19.19/6.67	
19.19/6.67	The pumping substitution is [X / n__g(X)].
19.19/6.67	
19.19/6.67	The result substitution is [ ].
19.19/6.67	
19.19/6.67	
19.19/6.67	
19.19/6.67	
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(8)
19.19/6.67	Complex Obligation (BEST)
19.19/6.67	
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(9)
19.19/6.67	Obligation:
19.19/6.67	Proved the lower bound n^1 for the following obligation:
19.19/6.67	
19.19/6.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.19/6.67	
19.19/6.67	
19.19/6.67	The TRS R consists of the following rules:
19.19/6.67	
19.19/6.67	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
19.19/6.67	   f(X) -> n__f(X)
19.19/6.67	   a -> n__a
19.19/6.67	   g(X) -> n__g(X)
19.19/6.67	   activate(n__f(X)) -> f(X)
19.19/6.67	   activate(n__a) -> a
19.19/6.67	   activate(n__g(X)) -> g(activate(X))
19.19/6.67	   activate(X) -> X
19.19/6.67	
19.19/6.67	S is empty.
19.19/6.67	Rewrite Strategy: FULL
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(10) LowerBoundPropagationProof (FINISHED)
19.19/6.67	Propagated lower bound.
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(11)
19.19/6.67	BOUNDS(n^1, INF)
19.19/6.67	
19.19/6.67	----------------------------------------
19.19/6.67	
19.19/6.67	(12)
19.19/6.67	Obligation:
19.19/6.67	Analyzing the following TRS for decreasing loops:
19.19/6.67	
19.19/6.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.19/6.67	
19.19/6.67	
19.19/6.67	The TRS R consists of the following rules:
19.19/6.67	
19.19/6.67	   f(n__f(n__a)) -> f(n__g(n__f(n__a)))
19.19/6.67	   f(X) -> n__f(X)
19.19/6.67	   a -> n__a
19.19/6.67	   g(X) -> n__g(X)
19.19/6.67	   activate(n__f(X)) -> f(X)
19.19/6.67	   activate(n__a) -> a
19.19/6.67	   activate(n__g(X)) -> g(activate(X))
19.19/6.67	   activate(X) -> X
19.19/6.67	
19.19/6.67	S is empty.
19.19/6.67	Rewrite Strategy: FULL
19.37/10.09	EOF