5.53/2.11	WORST_CASE(Omega(n^1), O(n^1))
5.53/2.12	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.53/2.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.53/2.12	
5.53/2.12	
5.53/2.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.53/2.12	
5.53/2.12	(0) CpxTRS
5.53/2.12	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
5.53/2.12	(2) CpxTRS
5.53/2.12	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
5.53/2.12	    (4) BOUNDS(1, n^1)
5.53/2.12	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.53/2.12	(6) TRS for Loop Detection
5.53/2.12	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.53/2.12	    (8) BEST
5.53/2.12	        (9) proven lower bound
5.53/2.12	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
5.53/2.12	            (11) BOUNDS(n^1, INF)
5.53/2.12	        (12) TRS for Loop Detection
5.53/2.12	
5.53/2.12	
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(0)
5.53/2.12	Obligation:
5.53/2.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.53/2.12	
5.53/2.12	
5.53/2.12	The TRS R consists of the following rules:
5.53/2.12	
5.53/2.12	   f(s(x)) -> s(s(f(p(s(x)))))
5.53/2.12	   f(0) -> 0
5.53/2.12	   p(s(x)) -> x
5.53/2.12	
5.53/2.12	S is empty.
5.53/2.12	Rewrite Strategy: INNERMOST
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(1) RelTrsToTrsProof (UPPER BOUND(ID))
5.53/2.12	transformed relative TRS to TRS
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(2)
5.53/2.12	Obligation:
5.53/2.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
5.53/2.12	
5.53/2.12	
5.53/2.12	The TRS R consists of the following rules:
5.53/2.12	
5.53/2.12	   f(s(x)) -> s(s(f(p(s(x)))))
5.53/2.12	   f(0) -> 0
5.53/2.12	   p(s(x)) -> x
5.53/2.12	
5.53/2.12	S is empty.
5.53/2.12	Rewrite Strategy: INNERMOST
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(3) CpxTrsMatchBoundsProof (FINISHED)
5.53/2.12	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
5.53/2.12	The certificate found is represented by the following graph.
5.53/2.12	
5.53/2.12	"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
5.53/2.12	{(1,2,[f_1|0, p_1|0, 0|1, s_1|1]), (1,3,[s_1|1]), (2,2,[s_1|0, 0|0]), (3,4,[s_1|1]), (4,5,[f_1|1]), (4,7,[s_1|2]), (4,2,[0|2]), (5,6,[p_1|1]), (5,2,[s_1|1, 0|1]), (6,2,[s_1|1]), (7,8,[s_1|2]), (8,9,[f_1|2]), (8,7,[s_1|2]), (8,2,[0|2]), (9,10,[p_1|2]), (9,2,[s_1|1, 0|1]), (10,2,[s_1|2])}"
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(4)
5.53/2.12	BOUNDS(1, n^1)
5.53/2.12	
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.53/2.12	Transformed a relative TRS into a decreasing-loop problem.
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(6)
5.53/2.12	Obligation:
5.53/2.12	Analyzing the following TRS for decreasing loops:
5.53/2.12	
5.53/2.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.53/2.12	
5.53/2.12	
5.53/2.12	The TRS R consists of the following rules:
5.53/2.12	
5.53/2.12	   f(s(x)) -> s(s(f(p(s(x)))))
5.53/2.12	   f(0) -> 0
5.53/2.12	   p(s(x)) -> x
5.53/2.12	
5.53/2.12	S is empty.
5.53/2.12	Rewrite Strategy: INNERMOST
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(7) DecreasingLoopProof (LOWER BOUND(ID))
5.53/2.12	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.53/2.12	
5.53/2.12	The rewrite sequence
5.53/2.12	
5.53/2.12	f(s(x)) ->^+ s(s(f(x)))
5.53/2.12	
5.53/2.12	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
5.53/2.12	
5.53/2.12	The pumping substitution is [x / s(x)].
5.53/2.12	
5.53/2.12	The result substitution is [ ].
5.53/2.12	
5.53/2.12	
5.53/2.12	
5.53/2.12	
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(8)
5.53/2.12	Complex Obligation (BEST)
5.53/2.12	
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(9)
5.53/2.12	Obligation:
5.53/2.12	Proved the lower bound n^1 for the following obligation:
5.53/2.12	
5.53/2.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.53/2.12	
5.53/2.12	
5.53/2.12	The TRS R consists of the following rules:
5.53/2.12	
5.53/2.12	   f(s(x)) -> s(s(f(p(s(x)))))
5.53/2.12	   f(0) -> 0
5.53/2.12	   p(s(x)) -> x
5.53/2.12	
5.53/2.12	S is empty.
5.53/2.12	Rewrite Strategy: INNERMOST
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(10) LowerBoundPropagationProof (FINISHED)
5.53/2.12	Propagated lower bound.
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(11)
5.53/2.12	BOUNDS(n^1, INF)
5.53/2.12	
5.53/2.12	----------------------------------------
5.53/2.12	
5.53/2.12	(12)
5.53/2.12	Obligation:
5.53/2.12	Analyzing the following TRS for decreasing loops:
5.53/2.12	
5.53/2.12	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.53/2.12	
5.53/2.12	
5.53/2.12	The TRS R consists of the following rules:
5.53/2.12	
5.53/2.12	   f(s(x)) -> s(s(f(p(s(x)))))
5.53/2.12	   f(0) -> 0
5.53/2.12	   p(s(x)) -> x
5.53/2.12	
5.53/2.12	S is empty.
5.53/2.12	Rewrite Strategy: INNERMOST
5.53/2.15	EOF