2.93/1.55	WORST_CASE(Omega(n^1), O(n^1))
2.93/1.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.93/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.93/1.55	
2.93/1.55	
2.93/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.93/1.55	
2.93/1.55	(0) CpxTRS
2.93/1.55	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
2.93/1.55	(2) CpxTRS
2.93/1.55	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
2.93/1.55	    (4) BOUNDS(1, n^1)
2.93/1.55	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.93/1.55	(6) TRS for Loop Detection
2.93/1.55	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.93/1.55	    (8) BEST
2.93/1.55	        (9) proven lower bound
2.93/1.55	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
2.93/1.55	            (11) BOUNDS(n^1, INF)
2.93/1.55	        (12) TRS for Loop Detection
2.93/1.55	
2.93/1.55	
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(0)
2.93/1.55	Obligation:
2.93/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.93/1.55	
2.93/1.55	
2.93/1.55	The TRS R consists of the following rules:
2.93/1.55	
2.93/1.55	   f(f(a)) -> f(g(n__f(n__a)))
2.93/1.55	   f(X) -> n__f(X)
2.93/1.55	   a -> n__a
2.93/1.55	   activate(n__f(X)) -> f(activate(X))
2.93/1.55	   activate(n__a) -> a
2.93/1.55	   activate(X) -> X
2.93/1.55	
2.93/1.55	S is empty.
2.93/1.55	Rewrite Strategy: FULL
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(1) RelTrsToTrsProof (UPPER BOUND(ID))
2.93/1.55	transformed relative TRS to TRS
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(2)
2.93/1.55	Obligation:
2.93/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
2.93/1.55	
2.93/1.55	
2.93/1.55	The TRS R consists of the following rules:
2.93/1.55	
2.93/1.55	   f(f(a)) -> f(g(n__f(n__a)))
2.93/1.55	   f(X) -> n__f(X)
2.93/1.55	   a -> n__a
2.93/1.55	   activate(n__f(X)) -> f(activate(X))
2.93/1.55	   activate(n__a) -> a
2.93/1.55	   activate(X) -> X
2.93/1.55	
2.93/1.55	S is empty.
2.93/1.55	Rewrite Strategy: FULL
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(3) CpxTrsMatchBoundsProof (FINISHED)
2.93/1.55	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
2.93/1.55	The certificate found is represented by the following graph.
2.93/1.55	
2.93/1.55	"[13, 14, 15, 16, 17, 18]
2.93/1.55	{(13,14,[f_1|0, a|0, activate_1|0, n__f_1|1, n__a|1, a|1, g_1|1, n__a|2]), (13,15,[f_1|1, n__f_1|2]), (13,16,[f_1|2, n__f_1|3]), (14,14,[g_1|0, n__f_1|0, n__a|0]), (15,14,[activate_1|1, n__f_1|1, a|1, n__a|1, g_1|1, n__a|2]), (15,15,[f_1|1, n__f_1|2]), (15,16,[f_1|2, n__f_1|3]), (16,17,[g_1|2]), (17,18,[n__f_1|2]), (18,14,[n__a|2])}"
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(4)
2.93/1.55	BOUNDS(1, n^1)
2.93/1.55	
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.93/1.55	Transformed a relative TRS into a decreasing-loop problem.
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(6)
2.93/1.55	Obligation:
2.93/1.55	Analyzing the following TRS for decreasing loops:
2.93/1.55	
2.93/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.93/1.55	
2.93/1.55	
2.93/1.55	The TRS R consists of the following rules:
2.93/1.55	
2.93/1.55	   f(f(a)) -> f(g(n__f(n__a)))
2.93/1.55	   f(X) -> n__f(X)
2.93/1.55	   a -> n__a
2.93/1.55	   activate(n__f(X)) -> f(activate(X))
2.93/1.55	   activate(n__a) -> a
2.93/1.55	   activate(X) -> X
2.93/1.55	
2.93/1.55	S is empty.
2.93/1.55	Rewrite Strategy: FULL
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(7) DecreasingLoopProof (LOWER BOUND(ID))
2.93/1.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.93/1.55	
2.93/1.55	The rewrite sequence
2.93/1.55	
2.93/1.55	activate(n__f(X)) ->^+ f(activate(X))
2.93/1.55	
2.93/1.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
2.93/1.55	
2.93/1.55	The pumping substitution is [X / n__f(X)].
2.93/1.55	
2.93/1.55	The result substitution is [ ].
2.93/1.55	
2.93/1.55	
2.93/1.55	
2.93/1.55	
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(8)
2.93/1.55	Complex Obligation (BEST)
2.93/1.55	
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(9)
2.93/1.55	Obligation:
2.93/1.55	Proved the lower bound n^1 for the following obligation:
2.93/1.55	
2.93/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.93/1.55	
2.93/1.55	
2.93/1.55	The TRS R consists of the following rules:
2.93/1.55	
2.93/1.55	   f(f(a)) -> f(g(n__f(n__a)))
2.93/1.55	   f(X) -> n__f(X)
2.93/1.55	   a -> n__a
2.93/1.55	   activate(n__f(X)) -> f(activate(X))
2.93/1.55	   activate(n__a) -> a
2.93/1.55	   activate(X) -> X
2.93/1.55	
2.93/1.55	S is empty.
2.93/1.55	Rewrite Strategy: FULL
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(10) LowerBoundPropagationProof (FINISHED)
2.93/1.55	Propagated lower bound.
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(11)
2.93/1.55	BOUNDS(n^1, INF)
2.93/1.55	
2.93/1.55	----------------------------------------
2.93/1.55	
2.93/1.55	(12)
2.93/1.55	Obligation:
2.93/1.55	Analyzing the following TRS for decreasing loops:
2.93/1.55	
2.93/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.93/1.55	
2.93/1.55	
2.93/1.55	The TRS R consists of the following rules:
2.93/1.55	
2.93/1.55	   f(f(a)) -> f(g(n__f(n__a)))
2.93/1.55	   f(X) -> n__f(X)
2.93/1.55	   a -> n__a
2.93/1.55	   activate(n__f(X)) -> f(activate(X))
2.93/1.55	   activate(n__a) -> a
2.93/1.55	   activate(X) -> X
2.93/1.55	
2.93/1.55	S is empty.
2.93/1.55	Rewrite Strategy: FULL
3.24/1.57	EOF