2.91/1.57	WORST_CASE(Omega(n^1), O(n^1))
3.25/1.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.25/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.25/1.57	
3.25/1.57	
3.25/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.25/1.57	
3.25/1.57	(0) CpxTRS
3.25/1.57	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.25/1.57	(2) CpxTRS
3.25/1.57	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.25/1.57	    (4) BOUNDS(1, n^1)
3.25/1.57	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.25/1.57	(6) TRS for Loop Detection
3.25/1.57	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.25/1.57	    (8) BEST
3.25/1.57	        (9) proven lower bound
3.25/1.57	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.25/1.57	            (11) BOUNDS(n^1, INF)
3.25/1.57	        (12) TRS for Loop Detection
3.25/1.57	
3.25/1.57	
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(0)
3.25/1.57	Obligation:
3.25/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.25/1.57	
3.25/1.57	
3.25/1.57	The TRS R consists of the following rules:
3.25/1.57	
3.25/1.57	   f(S(x), x2) -> f(x2, x)
3.25/1.57	   f(0, x2) -> 0
3.25/1.57	
3.25/1.57	S is empty.
3.25/1.57	Rewrite Strategy: INNERMOST
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.25/1.57	transformed relative TRS to TRS
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(2)
3.25/1.57	Obligation:
3.25/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.25/1.57	
3.25/1.57	
3.25/1.57	The TRS R consists of the following rules:
3.25/1.57	
3.25/1.57	   f(S(x), x2) -> f(x2, x)
3.25/1.57	   f(0, x2) -> 0
3.25/1.57	
3.25/1.57	S is empty.
3.25/1.57	Rewrite Strategy: INNERMOST
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.25/1.57	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. 
3.25/1.57	
3.25/1.57	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.25/1.57	final states : [1]
3.25/1.57	transitions: 
3.25/1.57	S0(0) -> 0
3.25/1.57	00() -> 0
3.25/1.57	f0(0, 0) -> 1
3.25/1.57	f1(0, 0) -> 1
3.25/1.57	01() -> 1
3.25/1.57	
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(4)
3.25/1.57	BOUNDS(1, n^1)
3.25/1.57	
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.25/1.57	Transformed a relative TRS into a decreasing-loop problem.
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(6)
3.25/1.57	Obligation:
3.25/1.57	Analyzing the following TRS for decreasing loops:
3.25/1.57	
3.25/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.25/1.57	
3.25/1.57	
3.25/1.57	The TRS R consists of the following rules:
3.25/1.57	
3.25/1.57	   f(S(x), x2) -> f(x2, x)
3.25/1.57	   f(0, x2) -> 0
3.25/1.57	
3.25/1.57	S is empty.
3.25/1.57	Rewrite Strategy: INNERMOST
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.25/1.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.25/1.57	
3.25/1.57	The rewrite sequence
3.25/1.57	
3.25/1.57	f(S(x), S(x1_0)) ->^+ f(x, x1_0)
3.25/1.57	
3.25/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.25/1.57	
3.25/1.57	The pumping substitution is [x / S(x), x1_0 / S(x1_0)].
3.25/1.57	
3.25/1.57	The result substitution is [ ].
3.25/1.57	
3.25/1.57	
3.25/1.57	
3.25/1.57	
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(8)
3.25/1.57	Complex Obligation (BEST)
3.25/1.57	
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(9)
3.25/1.57	Obligation:
3.25/1.57	Proved the lower bound n^1 for the following obligation:
3.25/1.57	
3.25/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.25/1.57	
3.25/1.57	
3.25/1.57	The TRS R consists of the following rules:
3.25/1.57	
3.25/1.57	   f(S(x), x2) -> f(x2, x)
3.25/1.57	   f(0, x2) -> 0
3.25/1.57	
3.25/1.57	S is empty.
3.25/1.57	Rewrite Strategy: INNERMOST
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(10) LowerBoundPropagationProof (FINISHED)
3.25/1.57	Propagated lower bound.
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(11)
3.25/1.57	BOUNDS(n^1, INF)
3.25/1.57	
3.25/1.57	----------------------------------------
3.25/1.57	
3.25/1.57	(12)
3.25/1.57	Obligation:
3.25/1.57	Analyzing the following TRS for decreasing loops:
3.25/1.57	
3.25/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.25/1.57	
3.25/1.57	
3.25/1.57	The TRS R consists of the following rules:
3.25/1.57	
3.25/1.57	   f(S(x), x2) -> f(x2, x)
3.25/1.57	   f(0, x2) -> 0
3.25/1.57	
3.25/1.57	S is empty.
3.25/1.57	Rewrite Strategy: INNERMOST
3.28/1.62	EOF