4.78/2.01	WORST_CASE(Omega(n^1), O(n^1))
4.78/2.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.78/2.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.78/2.01	
4.78/2.01	
4.78/2.01	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.78/2.01	
4.78/2.01	(0) CpxTRS
4.78/2.01	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.78/2.01	(2) CpxTRS
4.78/2.01	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
4.78/2.01	    (4) BOUNDS(1, n^1)
4.78/2.01	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.78/2.01	(6) TRS for Loop Detection
4.78/2.01	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.78/2.01	    (8) BEST
4.78/2.01	        (9) proven lower bound
4.78/2.01	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
4.78/2.01	            (11) BOUNDS(n^1, INF)
4.78/2.01	        (12) TRS for Loop Detection
4.78/2.01	
4.78/2.01	
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(0)
4.78/2.01	Obligation:
4.78/2.01	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.78/2.01	
4.78/2.01	
4.78/2.01	The TRS R consists of the following rules:
4.78/2.01	
4.78/2.01	   f(0) -> s(0)
4.78/2.01	   f(s(0)) -> s(0)
4.78/2.01	   f(s(s(x))) -> f(f(s(x)))
4.78/2.01	
4.78/2.01	S is empty.
4.78/2.01	Rewrite Strategy: FULL
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(1) RelTrsToTrsProof (UPPER BOUND(ID))
4.78/2.01	transformed relative TRS to TRS
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(2)
4.78/2.01	Obligation:
4.78/2.01	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.78/2.01	
4.78/2.01	
4.78/2.01	The TRS R consists of the following rules:
4.78/2.01	
4.78/2.01	   f(0) -> s(0)
4.78/2.01	   f(s(0)) -> s(0)
4.78/2.01	   f(s(s(x))) -> f(f(s(x)))
4.78/2.01	
4.78/2.01	S is empty.
4.78/2.01	Rewrite Strategy: FULL
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(3) CpxTrsMatchBoundsProof (FINISHED)
4.78/2.01	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
4.78/2.01	The certificate found is represented by the following graph.
4.78/2.01	
4.78/2.01	"[1, 2, 3, 4, 5, 6, 7]
4.78/2.01	{(1,2,[f_1|0]), (1,3,[s_1|1]), (1,4,[f_1|1]), (1,7,[s_1|2]), (2,2,[0|0, s_1|0]), (3,2,[0|1]), (4,5,[f_1|1]), (4,6,[s_1|1]), (4,4,[f_1|1]), (4,7,[s_1|2]), (5,2,[s_1|1]), (6,2,[0|1]), (7,2,[0|2])}"
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(4)
4.78/2.01	BOUNDS(1, n^1)
4.78/2.01	
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.78/2.01	Transformed a relative TRS into a decreasing-loop problem.
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(6)
4.78/2.01	Obligation:
4.78/2.01	Analyzing the following TRS for decreasing loops:
4.78/2.01	
4.78/2.01	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.78/2.01	
4.78/2.01	
4.78/2.01	The TRS R consists of the following rules:
4.78/2.01	
4.78/2.01	   f(0) -> s(0)
4.78/2.01	   f(s(0)) -> s(0)
4.78/2.01	   f(s(s(x))) -> f(f(s(x)))
4.78/2.01	
4.78/2.01	S is empty.
4.78/2.01	Rewrite Strategy: FULL
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(7) DecreasingLoopProof (LOWER BOUND(ID))
4.78/2.01	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.78/2.01	
4.78/2.01	The rewrite sequence
4.78/2.01	
4.78/2.01	f(s(s(x))) ->^+ f(f(s(x)))
4.78/2.01	
4.78/2.01	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.78/2.01	
4.78/2.01	The pumping substitution is [x / s(x)].
4.78/2.01	
4.78/2.01	The result substitution is [ ].
4.78/2.01	
4.78/2.01	
4.78/2.01	
4.78/2.01	
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(8)
4.78/2.01	Complex Obligation (BEST)
4.78/2.01	
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(9)
4.78/2.01	Obligation:
4.78/2.01	Proved the lower bound n^1 for the following obligation:
4.78/2.01	
4.78/2.01	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.78/2.01	
4.78/2.01	
4.78/2.01	The TRS R consists of the following rules:
4.78/2.01	
4.78/2.01	   f(0) -> s(0)
4.78/2.01	   f(s(0)) -> s(0)
4.78/2.01	   f(s(s(x))) -> f(f(s(x)))
4.78/2.01	
4.78/2.01	S is empty.
4.78/2.01	Rewrite Strategy: FULL
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(10) LowerBoundPropagationProof (FINISHED)
4.78/2.01	Propagated lower bound.
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(11)
4.78/2.01	BOUNDS(n^1, INF)
4.78/2.01	
4.78/2.01	----------------------------------------
4.78/2.01	
4.78/2.01	(12)
4.78/2.01	Obligation:
4.78/2.01	Analyzing the following TRS for decreasing loops:
4.78/2.01	
4.78/2.01	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
4.78/2.01	
4.78/2.01	
4.78/2.01	The TRS R consists of the following rules:
4.78/2.01	
4.78/2.01	   f(0) -> s(0)
4.78/2.01	   f(s(0)) -> s(0)
4.78/2.01	   f(s(s(x))) -> f(f(s(x)))
4.78/2.01	
4.78/2.01	S is empty.
4.78/2.01	Rewrite Strategy: FULL
5.00/2.06	EOF