12.50/4.29	WORST_CASE(Omega(n^1), O(n^1))
12.50/4.30	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
12.50/4.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.50/4.30	
12.50/4.30	
12.50/4.30	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.50/4.30	
12.50/4.30	(0) CpxTRS
12.50/4.30	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
12.50/4.30	(2) CpxTRS
12.50/4.30	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
12.50/4.30	    (4) BOUNDS(1, n^1)
12.50/4.30	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
12.50/4.30	(6) TRS for Loop Detection
12.50/4.30	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
12.50/4.30	    (8) BEST
12.50/4.30	        (9) proven lower bound
12.50/4.30	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
12.50/4.30	            (11) BOUNDS(n^1, INF)
12.50/4.30	        (12) TRS for Loop Detection
12.50/4.30	
12.50/4.30	
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(0)
12.50/4.30	Obligation:
12.50/4.30	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.50/4.30	
12.50/4.30	
12.50/4.30	The TRS R consists of the following rules:
12.50/4.30	
12.50/4.30	   a__f(f(a)) -> a__f(g(f(a)))
12.50/4.30	   mark(f(X)) -> a__f(X)
12.50/4.30	   mark(a) -> a
12.50/4.30	   mark(g(X)) -> g(mark(X))
12.50/4.30	   a__f(X) -> f(X)
12.50/4.30	
12.50/4.30	S is empty.
12.50/4.30	Rewrite Strategy: INNERMOST
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(1) RelTrsToTrsProof (UPPER BOUND(ID))
12.50/4.30	transformed relative TRS to TRS
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(2)
12.50/4.30	Obligation:
12.50/4.30	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
12.50/4.30	
12.50/4.30	
12.50/4.30	The TRS R consists of the following rules:
12.50/4.30	
12.50/4.30	   a__f(f(a)) -> a__f(g(f(a)))
12.50/4.30	   mark(f(X)) -> a__f(X)
12.50/4.30	   mark(a) -> a
12.50/4.30	   mark(g(X)) -> g(mark(X))
12.50/4.30	   a__f(X) -> f(X)
12.50/4.30	
12.50/4.30	S is empty.
12.50/4.30	Rewrite Strategy: INNERMOST
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(3) CpxTrsMatchBoundsProof (FINISHED)
12.50/4.30	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
12.50/4.30	The certificate found is represented by the following graph.
12.50/4.30	
12.50/4.30	"[1, 2, 3, 4, 5, 6]
12.50/4.30	{(1,2,[a__f_1|0, mark_1|0, f_1|1, a__f_1|1, a|1, f_1|2]), (1,3,[a__f_1|1, f_1|2]), (1,6,[g_1|1]), (2,2,[f_1|0, a|0, g_1|0]), (3,4,[g_1|1]), (4,5,[f_1|1]), (5,2,[a|1]), (6,2,[mark_1|1, a__f_1|1, a|1, f_1|2]), (6,6,[g_1|1]), (6,3,[a__f_1|1, f_1|2])}"
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(4)
12.50/4.30	BOUNDS(1, n^1)
12.50/4.30	
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
12.50/4.30	Transformed a relative TRS into a decreasing-loop problem.
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(6)
12.50/4.30	Obligation:
12.50/4.30	Analyzing the following TRS for decreasing loops:
12.50/4.30	
12.50/4.30	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.50/4.30	
12.50/4.30	
12.50/4.30	The TRS R consists of the following rules:
12.50/4.30	
12.50/4.30	   a__f(f(a)) -> a__f(g(f(a)))
12.50/4.30	   mark(f(X)) -> a__f(X)
12.50/4.30	   mark(a) -> a
12.50/4.30	   mark(g(X)) -> g(mark(X))
12.50/4.30	   a__f(X) -> f(X)
12.50/4.30	
12.50/4.30	S is empty.
12.50/4.30	Rewrite Strategy: INNERMOST
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(7) DecreasingLoopProof (LOWER BOUND(ID))
12.50/4.30	The following loop(s) give(s) rise to the lower bound Omega(n^1):
12.50/4.30	
12.50/4.30	The rewrite sequence
12.50/4.30	
12.50/4.30	mark(g(X)) ->^+ g(mark(X))
12.50/4.30	
12.50/4.30	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
12.50/4.30	
12.50/4.30	The pumping substitution is [X / g(X)].
12.50/4.30	
12.50/4.30	The result substitution is [ ].
12.50/4.30	
12.50/4.30	
12.50/4.30	
12.50/4.30	
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(8)
12.50/4.30	Complex Obligation (BEST)
12.50/4.30	
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(9)
12.50/4.30	Obligation:
12.50/4.30	Proved the lower bound n^1 for the following obligation:
12.50/4.30	
12.50/4.30	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.50/4.30	
12.50/4.30	
12.50/4.30	The TRS R consists of the following rules:
12.50/4.30	
12.50/4.30	   a__f(f(a)) -> a__f(g(f(a)))
12.50/4.30	   mark(f(X)) -> a__f(X)
12.50/4.30	   mark(a) -> a
12.50/4.30	   mark(g(X)) -> g(mark(X))
12.50/4.30	   a__f(X) -> f(X)
12.50/4.30	
12.50/4.30	S is empty.
12.50/4.30	Rewrite Strategy: INNERMOST
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(10) LowerBoundPropagationProof (FINISHED)
12.50/4.30	Propagated lower bound.
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(11)
12.50/4.30	BOUNDS(n^1, INF)
12.50/4.30	
12.50/4.30	----------------------------------------
12.50/4.30	
12.50/4.30	(12)
12.50/4.30	Obligation:
12.50/4.30	Analyzing the following TRS for decreasing loops:
12.50/4.30	
12.50/4.30	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
12.50/4.30	
12.50/4.30	
12.50/4.30	The TRS R consists of the following rules:
12.50/4.30	
12.50/4.30	   a__f(f(a)) -> a__f(g(f(a)))
12.50/4.30	   mark(f(X)) -> a__f(X)
12.50/4.30	   mark(a) -> a
12.50/4.30	   mark(g(X)) -> g(mark(X))
12.50/4.30	   a__f(X) -> f(X)
12.50/4.30	
12.50/4.30	S is empty.
12.50/4.30	Rewrite Strategy: INNERMOST
12.50/4.35	EOF