5.19/2.07	WORST_CASE(Omega(n^1), O(n^1))
5.19/2.07	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.19/2.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.19/2.07	
5.19/2.07	
5.19/2.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.19/2.07	
5.19/2.07	(0) CpxTRS
5.19/2.07	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
5.19/2.07	(2) CpxTRS
5.19/2.07	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
5.19/2.07	    (4) BOUNDS(1, n^1)
5.19/2.07	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.19/2.07	(6) TRS for Loop Detection
5.19/2.07	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.19/2.07	    (8) BEST
5.19/2.07	        (9) proven lower bound
5.19/2.07	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
5.19/2.07	            (11) BOUNDS(n^1, INF)
5.19/2.07	        (12) TRS for Loop Detection
5.19/2.07	
5.19/2.07	
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(0)
5.19/2.07	Obligation:
5.19/2.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.19/2.07	
5.19/2.07	
5.19/2.07	The TRS R consists of the following rules:
5.19/2.07	
5.19/2.07	   f(0) -> s(0)
5.19/2.07	   f(s(0)) -> s(0)
5.19/2.07	   f(s(s(x))) -> f(f(s(x)))
5.19/2.07	
5.19/2.07	S is empty.
5.19/2.07	Rewrite Strategy: INNERMOST
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(1) RelTrsToTrsProof (UPPER BOUND(ID))
5.19/2.07	transformed relative TRS to TRS
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(2)
5.19/2.07	Obligation:
5.19/2.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
5.19/2.07	
5.19/2.07	
5.19/2.07	The TRS R consists of the following rules:
5.19/2.07	
5.19/2.07	   f(0) -> s(0)
5.19/2.07	   f(s(0)) -> s(0)
5.19/2.07	   f(s(s(x))) -> f(f(s(x)))
5.19/2.07	
5.19/2.07	S is empty.
5.19/2.07	Rewrite Strategy: INNERMOST
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(3) CpxTrsMatchBoundsProof (FINISHED)
5.19/2.07	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
5.19/2.07	The certificate found is represented by the following graph.
5.19/2.07	
5.19/2.07	"[1, 2, 3, 4, 5, 6, 7]
5.19/2.07	{(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])}"
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(4)
5.19/2.07	BOUNDS(1, n^1)
5.19/2.07	
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.19/2.07	Transformed a relative TRS into a decreasing-loop problem.
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(6)
5.19/2.07	Obligation:
5.19/2.07	Analyzing the following TRS for decreasing loops:
5.19/2.07	
5.19/2.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.19/2.07	
5.19/2.07	
5.19/2.07	The TRS R consists of the following rules:
5.19/2.07	
5.19/2.07	   f(0) -> s(0)
5.19/2.07	   f(s(0)) -> s(0)
5.19/2.07	   f(s(s(x))) -> f(f(s(x)))
5.19/2.07	
5.19/2.07	S is empty.
5.19/2.07	Rewrite Strategy: INNERMOST
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(7) DecreasingLoopProof (LOWER BOUND(ID))
5.19/2.07	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.19/2.07	
5.19/2.07	The rewrite sequence
5.19/2.07	
5.19/2.07	f(s(s(x))) ->^+ f(f(s(x)))
5.19/2.07	
5.19/2.07	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.19/2.07	
5.19/2.07	The pumping substitution is [x / s(x)].
5.19/2.07	
5.19/2.07	The result substitution is [ ].
5.19/2.07	
5.19/2.07	
5.19/2.07	
5.19/2.07	
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(8)
5.19/2.07	Complex Obligation (BEST)
5.19/2.07	
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(9)
5.19/2.07	Obligation:
5.19/2.07	Proved the lower bound n^1 for the following obligation:
5.19/2.07	
5.19/2.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.19/2.07	
5.19/2.07	
5.19/2.07	The TRS R consists of the following rules:
5.19/2.07	
5.19/2.07	   f(0) -> s(0)
5.19/2.07	   f(s(0)) -> s(0)
5.19/2.07	   f(s(s(x))) -> f(f(s(x)))
5.19/2.07	
5.19/2.07	S is empty.
5.19/2.07	Rewrite Strategy: INNERMOST
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(10) LowerBoundPropagationProof (FINISHED)
5.19/2.07	Propagated lower bound.
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(11)
5.19/2.07	BOUNDS(n^1, INF)
5.19/2.07	
5.19/2.07	----------------------------------------
5.19/2.07	
5.19/2.07	(12)
5.19/2.07	Obligation:
5.19/2.07	Analyzing the following TRS for decreasing loops:
5.19/2.07	
5.19/2.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.19/2.07	
5.19/2.07	
5.19/2.07	The TRS R consists of the following rules:
5.19/2.07	
5.19/2.07	   f(0) -> s(0)
5.19/2.07	   f(s(0)) -> s(0)
5.19/2.07	   f(s(s(x))) -> f(f(s(x)))
5.19/2.07	
5.19/2.07	S is empty.
5.19/2.07	Rewrite Strategy: INNERMOST
5.19/2.10	EOF