3.54/1.65	WORST_CASE(Omega(n^1), O(n^1))
3.54/1.66	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.54/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.54/1.66	
3.54/1.66	
3.54/1.66	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.54/1.66	
3.54/1.66	(0) CpxTRS
3.54/1.66	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.54/1.66	(2) CpxTRS
3.54/1.66	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 35 ms]
3.54/1.66	    (4) BOUNDS(1, n^1)
3.54/1.66	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.54/1.66	(6) TRS for Loop Detection
3.54/1.66	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.54/1.66	    (8) BEST
3.54/1.66	        (9) proven lower bound
3.54/1.66	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.54/1.66	            (11) BOUNDS(n^1, INF)
3.54/1.66	        (12) TRS for Loop Detection
3.54/1.66	
3.54/1.66	
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(0)
3.54/1.66	Obligation:
3.54/1.66	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.54/1.66	
3.54/1.66	
3.54/1.66	The TRS R consists of the following rules:
3.54/1.66	
3.54/1.66	   +(0, y) -> y
3.54/1.66	   +(s(x), 0) -> s(x)
3.54/1.66	   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))
3.54/1.66	
3.54/1.66	S is empty.
3.54/1.66	Rewrite Strategy: INNERMOST
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.54/1.66	transformed relative TRS to TRS
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(2)
3.54/1.66	Obligation:
3.54/1.66	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.54/1.66	
3.54/1.66	
3.54/1.66	The TRS R consists of the following rules:
3.54/1.66	
3.54/1.66	   +(0, y) -> y
3.54/1.66	   +(s(x), 0) -> s(x)
3.54/1.66	   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))
3.54/1.66	
3.54/1.66	S is empty.
3.54/1.66	Rewrite Strategy: INNERMOST
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.54/1.66	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 3. 
3.54/1.66	
3.54/1.66	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.54/1.66	final states : [1]
3.54/1.66	transitions: 
3.54/1.66	00() -> 0
3.54/1.66	s0(0) -> 0
3.54/1.66	+0(0, 0) -> 1
3.54/1.66	s1(0) -> 1
3.54/1.66	s1(0) -> 3
3.54/1.66	01() -> 5
3.54/1.66	+1(0, 5) -> 4
3.54/1.66	+1(3, 4) -> 2
3.54/1.66	s1(2) -> 1
3.54/1.66	s1(0) -> 4
3.54/1.66	s2(0) -> 2
3.54/1.66	s2(0) -> 7
3.54/1.66	02() -> 9
3.54/1.66	+2(0, 9) -> 8
3.54/1.66	+2(7, 8) -> 6
3.54/1.66	s2(6) -> 2
3.54/1.66	s1(0) -> 8
3.54/1.66	s3(0) -> 6
3.54/1.66	s2(6) -> 6
3.54/1.66	0 -> 1
3.54/1.66	5 -> 4
3.54/1.66	9 -> 8
3.54/1.66	
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(4)
3.54/1.66	BOUNDS(1, n^1)
3.54/1.66	
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.54/1.66	Transformed a relative TRS into a decreasing-loop problem.
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(6)
3.54/1.66	Obligation:
3.54/1.66	Analyzing the following TRS for decreasing loops:
3.54/1.66	
3.54/1.66	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.54/1.66	
3.54/1.66	
3.54/1.66	The TRS R consists of the following rules:
3.54/1.66	
3.54/1.66	   +(0, y) -> y
3.54/1.66	   +(s(x), 0) -> s(x)
3.54/1.66	   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))
3.54/1.66	
3.54/1.66	S is empty.
3.54/1.66	Rewrite Strategy: INNERMOST
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.54/1.66	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.54/1.66	
3.54/1.66	The rewrite sequence
3.54/1.66	
3.54/1.66	+(s(x), s(s(x1_0))) ->^+ s(+(s(x), s(x1_0)))
3.54/1.66	
3.54/1.66	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.54/1.66	
3.54/1.66	The pumping substitution is [x1_0 / s(x1_0)].
3.54/1.66	
3.54/1.66	The result substitution is [ ].
3.54/1.66	
3.54/1.66	
3.54/1.66	
3.54/1.66	
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(8)
3.54/1.66	Complex Obligation (BEST)
3.54/1.66	
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(9)
3.54/1.66	Obligation:
3.54/1.66	Proved the lower bound n^1 for the following obligation:
3.54/1.66	
3.54/1.66	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.54/1.66	
3.54/1.66	
3.54/1.66	The TRS R consists of the following rules:
3.54/1.66	
3.54/1.66	   +(0, y) -> y
3.54/1.66	   +(s(x), 0) -> s(x)
3.54/1.66	   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))
3.54/1.66	
3.54/1.66	S is empty.
3.54/1.66	Rewrite Strategy: INNERMOST
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(10) LowerBoundPropagationProof (FINISHED)
3.54/1.66	Propagated lower bound.
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(11)
3.54/1.66	BOUNDS(n^1, INF)
3.54/1.66	
3.54/1.66	----------------------------------------
3.54/1.66	
3.54/1.66	(12)
3.54/1.66	Obligation:
3.54/1.66	Analyzing the following TRS for decreasing loops:
3.54/1.66	
3.54/1.66	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.54/1.66	
3.54/1.66	
3.54/1.66	The TRS R consists of the following rules:
3.54/1.66	
3.54/1.66	   +(0, y) -> y
3.54/1.66	   +(s(x), 0) -> s(x)
3.54/1.66	   +(s(x), s(y)) -> s(+(s(x), +(y, 0)))
3.54/1.66	
3.54/1.66	S is empty.
3.54/1.66	Rewrite Strategy: INNERMOST
3.71/1.68	EOF