18.10/5.79	WORST_CASE(Omega(n^1), O(n^1))
18.10/5.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
18.10/5.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.10/5.79	
18.10/5.79	
18.10/5.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.10/5.79	
18.10/5.79	(0) CpxTRS
18.10/5.79	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
18.10/5.79	(2) CpxTRS
18.10/5.79	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
18.10/5.79	    (4) BOUNDS(1, n^1)
18.10/5.79	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
18.10/5.79	(6) TRS for Loop Detection
18.10/5.79	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
18.10/5.79	    (8) BEST
18.10/5.79	        (9) proven lower bound
18.10/5.79	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
18.10/5.79	            (11) BOUNDS(n^1, INF)
18.10/5.79	        (12) TRS for Loop Detection
18.10/5.79	
18.10/5.79	
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(0)
18.10/5.79	Obligation:
18.10/5.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.10/5.79	
18.10/5.79	
18.10/5.79	The TRS R consists of the following rules:
18.10/5.79	
18.10/5.79	   a__f(f(a)) -> a__f(g(f(a)))
18.10/5.79	   mark(f(X)) -> a__f(X)
18.10/5.79	   mark(a) -> a
18.10/5.79	   mark(g(X)) -> g(mark(X))
18.10/5.79	   a__f(X) -> f(X)
18.10/5.79	
18.10/5.79	S is empty.
18.10/5.79	Rewrite Strategy: FULL
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(1) RelTrsToTrsProof (UPPER BOUND(ID))
18.10/5.79	transformed relative TRS to TRS
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(2)
18.10/5.79	Obligation:
18.10/5.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
18.10/5.79	
18.10/5.79	
18.10/5.79	The TRS R consists of the following rules:
18.10/5.79	
18.10/5.79	   a__f(f(a)) -> a__f(g(f(a)))
18.10/5.79	   mark(f(X)) -> a__f(X)
18.10/5.79	   mark(a) -> a
18.10/5.79	   mark(g(X)) -> g(mark(X))
18.10/5.79	   a__f(X) -> f(X)
18.10/5.79	
18.10/5.79	S is empty.
18.10/5.79	Rewrite Strategy: FULL
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(3) CpxTrsMatchBoundsProof (FINISHED)
18.10/5.79	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
18.10/5.79	The certificate found is represented by the following graph.
18.10/5.79	
18.10/5.79	"[1, 2, 7, 8, 9, 10]
18.10/5.79	{(1,2,[a__f_1|0, mark_1|0, f_1|1, a__f_1|1, a|1, f_1|2]), (1,7,[a__f_1|1, f_1|2]), (1,10,[g_1|1]), (2,2,[f_1|0, a|0, g_1|0]), (7,8,[g_1|1]), (8,9,[f_1|1]), (9,2,[a|1]), (10,2,[mark_1|1, a__f_1|1, a|1, f_1|2]), (10,10,[g_1|1]), (10,7,[a__f_1|1, f_1|2])}"
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(4)
18.10/5.79	BOUNDS(1, n^1)
18.10/5.79	
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
18.10/5.79	Transformed a relative TRS into a decreasing-loop problem.
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(6)
18.10/5.79	Obligation:
18.10/5.79	Analyzing the following TRS for decreasing loops:
18.10/5.79	
18.10/5.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.10/5.79	
18.10/5.79	
18.10/5.79	The TRS R consists of the following rules:
18.10/5.79	
18.10/5.79	   a__f(f(a)) -> a__f(g(f(a)))
18.10/5.79	   mark(f(X)) -> a__f(X)
18.10/5.79	   mark(a) -> a
18.10/5.79	   mark(g(X)) -> g(mark(X))
18.10/5.79	   a__f(X) -> f(X)
18.10/5.79	
18.10/5.79	S is empty.
18.10/5.79	Rewrite Strategy: FULL
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(7) DecreasingLoopProof (LOWER BOUND(ID))
18.10/5.79	The following loop(s) give(s) rise to the lower bound Omega(n^1):
18.10/5.79	
18.10/5.79	The rewrite sequence
18.10/5.79	
18.10/5.79	mark(g(X)) ->^+ g(mark(X))
18.10/5.79	
18.10/5.79	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
18.10/5.79	
18.10/5.79	The pumping substitution is [X / g(X)].
18.10/5.79	
18.10/5.79	The result substitution is [ ].
18.10/5.79	
18.10/5.79	
18.10/5.79	
18.10/5.79	
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(8)
18.10/5.79	Complex Obligation (BEST)
18.10/5.79	
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(9)
18.10/5.79	Obligation:
18.10/5.79	Proved the lower bound n^1 for the following obligation:
18.10/5.79	
18.10/5.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.10/5.79	
18.10/5.79	
18.10/5.79	The TRS R consists of the following rules:
18.10/5.79	
18.10/5.79	   a__f(f(a)) -> a__f(g(f(a)))
18.10/5.79	   mark(f(X)) -> a__f(X)
18.10/5.79	   mark(a) -> a
18.10/5.79	   mark(g(X)) -> g(mark(X))
18.10/5.79	   a__f(X) -> f(X)
18.10/5.79	
18.10/5.79	S is empty.
18.10/5.79	Rewrite Strategy: FULL
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(10) LowerBoundPropagationProof (FINISHED)
18.10/5.79	Propagated lower bound.
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(11)
18.10/5.79	BOUNDS(n^1, INF)
18.10/5.79	
18.10/5.79	----------------------------------------
18.10/5.79	
18.10/5.79	(12)
18.10/5.79	Obligation:
18.10/5.79	Analyzing the following TRS for decreasing loops:
18.10/5.79	
18.10/5.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.10/5.79	
18.10/5.79	
18.10/5.79	The TRS R consists of the following rules:
18.10/5.79	
18.10/5.79	   a__f(f(a)) -> a__f(g(f(a)))
18.10/5.79	   mark(f(X)) -> a__f(X)
18.10/5.79	   mark(a) -> a
18.10/5.79	   mark(g(X)) -> g(mark(X))
18.10/5.79	   a__f(X) -> f(X)
18.10/5.79	
18.10/5.79	S is empty.
18.10/5.79	Rewrite Strategy: FULL
18.28/5.84	EOF