4.72/2.03	WORST_CASE(Omega(n^1), O(n^1))
4.72/2.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.72/2.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.72/2.03	
4.72/2.03	
4.72/2.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.72/2.03	
4.72/2.03	(0) CpxTRS
4.72/2.03	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.72/2.03	(2) CpxTRS
4.72/2.03	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
4.72/2.03	    (4) BOUNDS(1, n^1)
4.72/2.03	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.72/2.03	(6) TRS for Loop Detection
4.72/2.03	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.72/2.03	    (8) BEST
4.72/2.03	        (9) proven lower bound
4.72/2.03	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
4.72/2.03	            (11) BOUNDS(n^1, INF)
4.72/2.03	        (12) TRS for Loop Detection
4.72/2.03	
4.72/2.03	
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(0)
4.72/2.03	Obligation:
4.72/2.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.72/2.03	
4.72/2.03	
4.72/2.03	The TRS R consists of the following rules:
4.72/2.03	
4.72/2.03	   a__f(X) -> g(h(f(X)))
4.72/2.03	   mark(f(X)) -> a__f(mark(X))
4.72/2.03	   mark(g(X)) -> g(X)
4.72/2.03	   mark(h(X)) -> h(mark(X))
4.72/2.03	   a__f(X) -> f(X)
4.72/2.03	
4.72/2.03	S is empty.
4.72/2.03	Rewrite Strategy: INNERMOST
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(1) RelTrsToTrsProof (UPPER BOUND(ID))
4.72/2.03	transformed relative TRS to TRS
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(2)
4.72/2.03	Obligation:
4.72/2.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.72/2.03	
4.72/2.03	
4.72/2.03	The TRS R consists of the following rules:
4.72/2.03	
4.72/2.03	   a__f(X) -> g(h(f(X)))
4.72/2.03	   mark(f(X)) -> a__f(mark(X))
4.72/2.03	   mark(g(X)) -> g(X)
4.72/2.03	   mark(h(X)) -> h(mark(X))
4.72/2.03	   a__f(X) -> f(X)
4.72/2.03	
4.72/2.03	S is empty.
4.72/2.03	Rewrite Strategy: INNERMOST
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(3) CpxTrsMatchBoundsProof (FINISHED)
4.72/2.03	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
4.72/2.03	The certificate found is represented by the following graph.
4.72/2.03	
4.72/2.03	"[1, 2, 3, 4, 5, 6, 7, 8]
4.72/2.03	{(1,2,[a__f_1|0, mark_1|0, f_1|1, g_1|1]), (1,3,[g_1|1]), (1,5,[a__f_1|1, f_1|2]), (1,6,[h_1|1]), (1,7,[g_1|2]), (2,2,[g_1|0, h_1|0, f_1|0]), (3,4,[h_1|1]), (4,2,[f_1|1]), (5,2,[mark_1|1, g_1|1]), (5,5,[a__f_1|1, f_1|2]), (5,6,[h_1|1]), (5,7,[g_1|2]), (6,2,[mark_1|1, g_1|1]), (6,5,[a__f_1|1, f_1|2]), (6,6,[h_1|1]), (6,7,[g_1|2]), (7,8,[h_1|2]), (8,5,[f_1|2])}"
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(4)
4.72/2.03	BOUNDS(1, n^1)
4.72/2.03	
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.72/2.03	Transformed a relative TRS into a decreasing-loop problem.
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(6)
4.72/2.03	Obligation:
4.72/2.03	Analyzing the following TRS for decreasing loops:
4.72/2.03	
4.72/2.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.72/2.03	
4.72/2.03	
4.72/2.03	The TRS R consists of the following rules:
4.72/2.03	
4.72/2.03	   a__f(X) -> g(h(f(X)))
4.72/2.03	   mark(f(X)) -> a__f(mark(X))
4.72/2.03	   mark(g(X)) -> g(X)
4.72/2.03	   mark(h(X)) -> h(mark(X))
4.72/2.03	   a__f(X) -> f(X)
4.72/2.03	
4.72/2.03	S is empty.
4.72/2.03	Rewrite Strategy: INNERMOST
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(7) DecreasingLoopProof (LOWER BOUND(ID))
4.72/2.03	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.72/2.03	
4.72/2.03	The rewrite sequence
4.72/2.03	
4.72/2.03	mark(f(X)) ->^+ a__f(mark(X))
4.72/2.03	
4.72/2.03	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.72/2.03	
4.72/2.03	The pumping substitution is [X / f(X)].
4.72/2.03	
4.72/2.03	The result substitution is [ ].
4.72/2.03	
4.72/2.03	
4.72/2.03	
4.72/2.03	
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(8)
4.72/2.03	Complex Obligation (BEST)
4.72/2.03	
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(9)
4.72/2.03	Obligation:
4.72/2.03	Proved the lower bound n^1 for the following obligation:
4.72/2.03	
4.72/2.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.72/2.03	
4.72/2.03	
4.72/2.03	The TRS R consists of the following rules:
4.72/2.03	
4.72/2.03	   a__f(X) -> g(h(f(X)))
4.72/2.03	   mark(f(X)) -> a__f(mark(X))
4.72/2.03	   mark(g(X)) -> g(X)
4.72/2.03	   mark(h(X)) -> h(mark(X))
4.72/2.03	   a__f(X) -> f(X)
4.72/2.03	
4.72/2.03	S is empty.
4.72/2.03	Rewrite Strategy: INNERMOST
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(10) LowerBoundPropagationProof (FINISHED)
4.72/2.03	Propagated lower bound.
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(11)
4.72/2.03	BOUNDS(n^1, INF)
4.72/2.03	
4.72/2.03	----------------------------------------
4.72/2.03	
4.72/2.03	(12)
4.72/2.03	Obligation:
4.72/2.03	Analyzing the following TRS for decreasing loops:
4.72/2.03	
4.72/2.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.72/2.03	
4.72/2.03	
4.72/2.03	The TRS R consists of the following rules:
4.72/2.03	
4.72/2.03	   a__f(X) -> g(h(f(X)))
4.72/2.03	   mark(f(X)) -> a__f(mark(X))
4.72/2.03	   mark(g(X)) -> g(X)
4.72/2.03	   mark(h(X)) -> h(mark(X))
4.72/2.03	   a__f(X) -> f(X)
4.72/2.03	
4.72/2.03	S is empty.
4.72/2.03	Rewrite Strategy: INNERMOST
5.02/2.07	EOF