3.29/1.66	WORST_CASE(Omega(n^1), O(n^1))
3.29/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.29/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.29/1.67	
3.29/1.67	
3.29/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.67	
3.29/1.67	(0) CpxTRS
3.29/1.67	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.29/1.67	(2) CpxTRS
3.29/1.67	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
3.29/1.67	    (4) BOUNDS(1, n^1)
3.29/1.67	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.29/1.67	(6) TRS for Loop Detection
3.29/1.67	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.29/1.67	    (8) BEST
3.29/1.67	        (9) proven lower bound
3.29/1.67	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.29/1.67	            (11) BOUNDS(n^1, INF)
3.29/1.67	        (12) TRS for Loop Detection
3.29/1.67	
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(0)
3.29/1.67	Obligation:
3.29/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.67	
3.29/1.67	
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
3.29/1.67	   f(X) -> n__f(X)
3.29/1.67	   g(X) -> n__g(X)
3.29/1.67	   a -> n__a
3.29/1.67	   activate(n__f(X)) -> f(activate(X))
3.29/1.67	   activate(n__g(X)) -> g(activate(X))
3.29/1.67	   activate(n__a) -> a
3.29/1.67	   activate(X) -> X
3.29/1.67	
3.29/1.67	S is empty.
3.29/1.67	Rewrite Strategy: FULL
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.29/1.67	transformed relative TRS to TRS
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(2)
3.29/1.67	Obligation:
3.29/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.29/1.67	
3.29/1.67	
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
3.29/1.67	   f(X) -> n__f(X)
3.29/1.67	   g(X) -> n__g(X)
3.29/1.67	   a -> n__a
3.29/1.67	   activate(n__f(X)) -> f(activate(X))
3.29/1.67	   activate(n__g(X)) -> g(activate(X))
3.29/1.67	   activate(n__a) -> a
3.29/1.67	   activate(X) -> X
3.29/1.67	
3.29/1.67	S is empty.
3.29/1.67	Rewrite Strategy: FULL
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(3) CpxTrsMatchBoundsProof (FINISHED)
3.29/1.67	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
3.29/1.67	The certificate found is represented by the following graph.
3.29/1.67	
3.29/1.67	"[21, 22, 23, 24, 25, 26, 27, 28]
3.29/1.67	{(21,22,[f_1|0, g_1|0, a|0, activate_1|0, n__f_1|1, n__g_1|1, n__a|1, a|1, c_1|1, n__a|2]), (21,23,[f_1|1, n__f_1|2]), (21,24,[g_1|1, n__g_1|2]), (21,25,[c_1|2]), (22,22,[c_1|0, n__f_1|0, n__g_1|0, n__a|0]), (23,22,[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]), (23,23,[f_1|1, n__f_1|2]), (23,24,[g_1|1, n__g_1|2]), (23,25,[c_1|2]), (24,22,[activate_1|1, n__f_1|1, n__g_1|1, a|1, n__a|1, c_1|1, n__a|2]), (24,23,[f_1|1, n__f_1|2]), (24,24,[g_1|1, n__g_1|2]), (24,25,[c_1|2]), (25,26,[n__f_1|2]), (26,27,[n__g_1|2]), (27,28,[n__f_1|2]), (28,22,[n__a|2])}"
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(4)
3.29/1.67	BOUNDS(1, n^1)
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.29/1.67	Transformed a relative TRS into a decreasing-loop problem.
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(6)
3.29/1.67	Obligation:
3.29/1.67	Analyzing the following TRS for decreasing loops:
3.29/1.67	
3.29/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.67	
3.29/1.67	
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
3.29/1.67	   f(X) -> n__f(X)
3.29/1.67	   g(X) -> n__g(X)
3.29/1.67	   a -> n__a
3.29/1.67	   activate(n__f(X)) -> f(activate(X))
3.29/1.67	   activate(n__g(X)) -> g(activate(X))
3.29/1.67	   activate(n__a) -> a
3.29/1.67	   activate(X) -> X
3.29/1.67	
3.29/1.67	S is empty.
3.29/1.67	Rewrite Strategy: FULL
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.29/1.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.29/1.67	
3.29/1.67	The rewrite sequence
3.29/1.67	
3.29/1.67	activate(n__f(X)) ->^+ f(activate(X))
3.29/1.67	
3.29/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.29/1.67	
3.29/1.67	The pumping substitution is [X / n__f(X)].
3.29/1.67	
3.29/1.67	The result substitution is [ ].
3.29/1.67	
3.29/1.67	
3.29/1.67	
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(8)
3.29/1.67	Complex Obligation (BEST)
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(9)
3.29/1.67	Obligation:
3.29/1.67	Proved the lower bound n^1 for the following obligation:
3.29/1.67	
3.29/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.67	
3.29/1.67	
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
3.29/1.67	   f(X) -> n__f(X)
3.29/1.67	   g(X) -> n__g(X)
3.29/1.67	   a -> n__a
3.29/1.67	   activate(n__f(X)) -> f(activate(X))
3.29/1.67	   activate(n__g(X)) -> g(activate(X))
3.29/1.67	   activate(n__a) -> a
3.29/1.67	   activate(X) -> X
3.29/1.67	
3.29/1.67	S is empty.
3.29/1.67	Rewrite Strategy: FULL
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(10) LowerBoundPropagationProof (FINISHED)
3.29/1.67	Propagated lower bound.
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(11)
3.29/1.67	BOUNDS(n^1, INF)
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(12)
3.29/1.67	Obligation:
3.29/1.67	Analyzing the following TRS for decreasing loops:
3.29/1.67	
3.29/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.67	
3.29/1.67	
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
3.29/1.67	   f(X) -> n__f(X)
3.29/1.67	   g(X) -> n__g(X)
3.29/1.67	   a -> n__a
3.29/1.67	   activate(n__f(X)) -> f(activate(X))
3.29/1.67	   activate(n__g(X)) -> g(activate(X))
3.29/1.67	   activate(n__a) -> a
3.29/1.67	   activate(X) -> X
3.29/1.67	
3.29/1.67	S is empty.
3.29/1.67	Rewrite Strategy: FULL
3.44/1.69	EOF