5.66/2.19	WORST_CASE(Omega(n^1), O(n^1))
5.66/2.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.66/2.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.66/2.20	
5.66/2.20	
5.66/2.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.66/2.20	
5.66/2.20	(0) CpxTRS
5.66/2.20	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
5.66/2.20	(2) CpxTRS
5.66/2.20	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
5.66/2.20	    (4) BOUNDS(1, n^1)
5.66/2.20	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.66/2.20	(6) TRS for Loop Detection
5.66/2.20	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.66/2.20	    (8) BEST
5.66/2.20	        (9) proven lower bound
5.66/2.20	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
5.66/2.20	            (11) BOUNDS(n^1, INF)
5.66/2.20	        (12) TRS for Loop Detection
5.66/2.20	
5.66/2.20	
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(0)
5.66/2.20	Obligation:
5.66/2.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.66/2.20	
5.66/2.20	
5.66/2.20	The TRS R consists of the following rules:
5.66/2.20	
5.66/2.20	   f(x, a) -> x
5.66/2.20	   f(x, g(y)) -> f(g(x), y)
5.66/2.20	
5.66/2.20	S is empty.
5.66/2.20	Rewrite Strategy: FULL
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(1) RelTrsToTrsProof (UPPER BOUND(ID))
5.66/2.20	transformed relative TRS to TRS
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(2)
5.66/2.20	Obligation:
5.66/2.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
5.66/2.20	
5.66/2.20	
5.66/2.20	The TRS R consists of the following rules:
5.66/2.20	
5.66/2.20	   f(x, a) -> x
5.66/2.20	   f(x, g(y)) -> f(g(x), y)
5.66/2.20	
5.66/2.20	S is empty.
5.66/2.20	Rewrite Strategy: FULL
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(3) CpxTrsMatchBoundsTAProof (FINISHED)
5.66/2.20	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. 
5.66/2.20	
5.66/2.20	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
5.66/2.20	final states : [1]
5.66/2.20	transitions: 
5.66/2.20	a0() -> 0
5.66/2.20	g0(0) -> 0
5.66/2.20	f0(0, 0) -> 1
5.66/2.20	g1(0) -> 2
5.66/2.20	f1(2, 0) -> 1
5.66/2.20	g1(2) -> 2
5.66/2.20	0 -> 1
5.66/2.20	2 -> 1
5.66/2.20	
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(4)
5.66/2.20	BOUNDS(1, n^1)
5.66/2.20	
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.66/2.20	Transformed a relative TRS into a decreasing-loop problem.
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(6)
5.66/2.20	Obligation:
5.66/2.20	Analyzing the following TRS for decreasing loops:
5.66/2.20	
5.66/2.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.66/2.20	
5.66/2.20	
5.66/2.20	The TRS R consists of the following rules:
5.66/2.20	
5.66/2.20	   f(x, a) -> x
5.66/2.20	   f(x, g(y)) -> f(g(x), y)
5.66/2.20	
5.66/2.20	S is empty.
5.66/2.20	Rewrite Strategy: FULL
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(7) DecreasingLoopProof (LOWER BOUND(ID))
5.66/2.20	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.66/2.20	
5.66/2.20	The rewrite sequence
5.66/2.20	
5.66/2.20	f(x, g(y)) ->^+ f(g(x), y)
5.66/2.20	
5.66/2.20	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
5.66/2.20	
5.66/2.20	The pumping substitution is [y / g(y)].
5.66/2.20	
5.66/2.20	The result substitution is [x / g(x)].
5.66/2.20	
5.66/2.20	
5.66/2.20	
5.66/2.20	
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(8)
5.66/2.20	Complex Obligation (BEST)
5.66/2.20	
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(9)
5.66/2.20	Obligation:
5.66/2.20	Proved the lower bound n^1 for the following obligation:
5.66/2.20	
5.66/2.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.66/2.20	
5.66/2.20	
5.66/2.20	The TRS R consists of the following rules:
5.66/2.20	
5.66/2.20	   f(x, a) -> x
5.66/2.20	   f(x, g(y)) -> f(g(x), y)
5.66/2.20	
5.66/2.20	S is empty.
5.66/2.20	Rewrite Strategy: FULL
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(10) LowerBoundPropagationProof (FINISHED)
5.66/2.20	Propagated lower bound.
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(11)
5.66/2.20	BOUNDS(n^1, INF)
5.66/2.20	
5.66/2.20	----------------------------------------
5.66/2.20	
5.66/2.20	(12)
5.66/2.20	Obligation:
5.66/2.20	Analyzing the following TRS for decreasing loops:
5.66/2.20	
5.66/2.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
5.66/2.20	
5.66/2.20	
5.66/2.20	The TRS R consists of the following rules:
5.66/2.20	
5.66/2.20	   f(x, a) -> x
5.66/2.20	   f(x, g(y)) -> f(g(x), y)
5.66/2.20	
5.66/2.20	S is empty.
5.66/2.20	Rewrite Strategy: FULL
5.70/2.24	EOF