3.43/1.63	WORST_CASE(Omega(n^1), O(n^1))
3.43/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.64	
3.43/1.64	
3.43/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	(0) CpxTRS
3.43/1.64	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.43/1.64	(2) CpxTRS
3.43/1.64	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.43/1.64	    (4) BOUNDS(1, n^1)
3.43/1.64	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.43/1.64	(6) TRS for Loop Detection
3.43/1.64	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.43/1.64	    (8) BEST
3.43/1.64	        (9) proven lower bound
3.43/1.64	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.43/1.64	            (11) BOUNDS(n^1, INF)
3.43/1.64	        (12) TRS for Loop Detection
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(0)
3.43/1.64	Obligation:
3.43/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   +(0, y) -> y
3.43/1.64	   +(s(x), y) -> s(+(x, y))
3.43/1.64	   -(0, y) -> 0
3.43/1.64	   -(x, 0) -> x
3.43/1.64	   -(s(x), s(y)) -> -(x, y)
3.43/1.64	
3.43/1.64	S is empty.
3.43/1.64	Rewrite Strategy: FULL
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.43/1.65	transformed relative TRS to TRS
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(2)
3.43/1.65	Obligation:
3.43/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.43/1.65	
3.43/1.65	
3.43/1.65	The TRS R consists of the following rules:
3.43/1.65	
3.43/1.65	   +(0, y) -> y
3.43/1.65	   +(s(x), y) -> s(+(x, y))
3.43/1.65	   -(0, y) -> 0
3.43/1.65	   -(x, 0) -> x
3.43/1.65	   -(s(x), s(y)) -> -(x, y)
3.43/1.65	
3.43/1.65	S is empty.
3.43/1.65	Rewrite Strategy: FULL
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.43/1.65	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 1. 
3.43/1.65	
3.43/1.65	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.43/1.65	final states : [1, 2]
3.43/1.65	transitions: 
3.43/1.65	00() -> 0
3.43/1.65	s0(0) -> 0
3.43/1.65	+0(0, 0) -> 1
3.43/1.65	-0(0, 0) -> 2
3.43/1.65	+1(0, 0) -> 3
3.43/1.65	s1(3) -> 1
3.43/1.65	01() -> 2
3.43/1.65	-1(0, 0) -> 2
3.43/1.65	s1(3) -> 3
3.43/1.65	0 -> 1
3.43/1.65	0 -> 2
3.43/1.65	0 -> 3
3.43/1.65	
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(4)
3.43/1.65	BOUNDS(1, n^1)
3.43/1.65	
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.43/1.65	Transformed a relative TRS into a decreasing-loop problem.
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(6)
3.43/1.65	Obligation:
3.43/1.65	Analyzing the following TRS for decreasing loops:
3.43/1.65	
3.43/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.65	
3.43/1.65	
3.43/1.65	The TRS R consists of the following rules:
3.43/1.65	
3.43/1.65	   +(0, y) -> y
3.43/1.65	   +(s(x), y) -> s(+(x, y))
3.43/1.65	   -(0, y) -> 0
3.43/1.65	   -(x, 0) -> x
3.43/1.65	   -(s(x), s(y)) -> -(x, y)
3.43/1.65	
3.43/1.65	S is empty.
3.43/1.65	Rewrite Strategy: FULL
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.43/1.65	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.43/1.65	
3.43/1.65	The rewrite sequence
3.43/1.65	
3.43/1.65	-(s(x), s(y)) ->^+ -(x, y)
3.43/1.65	
3.43/1.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.43/1.65	
3.43/1.65	The pumping substitution is [x / s(x), y / s(y)].
3.43/1.65	
3.43/1.65	The result substitution is [ ].
3.43/1.65	
3.43/1.65	
3.43/1.65	
3.43/1.65	
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(8)
3.43/1.65	Complex Obligation (BEST)
3.43/1.65	
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(9)
3.43/1.65	Obligation:
3.43/1.65	Proved the lower bound n^1 for the following obligation:
3.43/1.65	
3.43/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.65	
3.43/1.65	
3.43/1.65	The TRS R consists of the following rules:
3.43/1.65	
3.43/1.65	   +(0, y) -> y
3.43/1.65	   +(s(x), y) -> s(+(x, y))
3.43/1.65	   -(0, y) -> 0
3.43/1.65	   -(x, 0) -> x
3.43/1.65	   -(s(x), s(y)) -> -(x, y)
3.43/1.65	
3.43/1.65	S is empty.
3.43/1.65	Rewrite Strategy: FULL
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(10) LowerBoundPropagationProof (FINISHED)
3.43/1.65	Propagated lower bound.
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(11)
3.43/1.65	BOUNDS(n^1, INF)
3.43/1.65	
3.43/1.65	----------------------------------------
3.43/1.65	
3.43/1.65	(12)
3.43/1.65	Obligation:
3.43/1.65	Analyzing the following TRS for decreasing loops:
3.43/1.65	
3.43/1.65	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.65	
3.43/1.65	
3.43/1.65	The TRS R consists of the following rules:
3.43/1.65	
3.43/1.65	   +(0, y) -> y
3.43/1.65	   +(s(x), y) -> s(+(x, y))
3.43/1.65	   -(0, y) -> 0
3.43/1.65	   -(x, 0) -> x
3.43/1.65	   -(s(x), s(y)) -> -(x, y)
3.43/1.65	
3.43/1.65	S is empty.
3.43/1.65	Rewrite Strategy: FULL
3.43/1.67	EOF