959.15/291.54	WORST_CASE(Omega(n^1), ?)
960.26/291.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
960.26/291.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
960.26/291.75	
960.26/291.75	
960.26/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
960.26/291.75	
960.26/291.75	(0) CpxTRS
960.26/291.75	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
960.26/291.75	(2) TRS for Loop Detection
960.26/291.75	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
960.26/291.75	(4) BEST
960.26/291.75	    (5) proven lower bound
960.26/291.75	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
960.26/291.75	        (7) BOUNDS(n^1, INF)
960.26/291.75	    (8) TRS for Loop Detection
960.26/291.75	
960.26/291.75	
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(0)
960.26/291.75	Obligation:
960.26/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
960.26/291.75	
960.26/291.75	
960.26/291.75	The TRS R consists of the following rules:
960.26/291.75	
960.26/291.75	   f(x, a(b(y))) -> f(a(b(b(x))), y)
960.26/291.75	   f(a(x), y) -> f(x, a(y))
960.26/291.75	   f(b(x), y) -> f(x, b(y))
960.26/291.75	
960.26/291.75	S is empty.
960.26/291.75	Rewrite Strategy: FULL
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
960.26/291.75	Transformed a relative TRS into a decreasing-loop problem.
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(2)
960.26/291.75	Obligation:
960.26/291.75	Analyzing the following TRS for decreasing loops:
960.26/291.75	
960.26/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
960.26/291.75	
960.26/291.75	
960.26/291.75	The TRS R consists of the following rules:
960.26/291.75	
960.26/291.75	   f(x, a(b(y))) -> f(a(b(b(x))), y)
960.26/291.75	   f(a(x), y) -> f(x, a(y))
960.26/291.75	   f(b(x), y) -> f(x, b(y))
960.26/291.75	
960.26/291.75	S is empty.
960.26/291.75	Rewrite Strategy: FULL
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(3) DecreasingLoopProof (LOWER BOUND(ID))
960.26/291.75	The following loop(s) give(s) rise to the lower bound Omega(n^1):
960.26/291.75	
960.26/291.75	The rewrite sequence
960.26/291.75	
960.26/291.75	f(x, a(b(y))) ->^+ f(a(b(b(x))), y)
960.26/291.75	
960.26/291.75	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
960.26/291.75	
960.26/291.75	The pumping substitution is [y / a(b(y))].
960.26/291.75	
960.26/291.75	The result substitution is [x / a(b(b(x)))].
960.26/291.75	
960.26/291.75	
960.26/291.75	
960.26/291.75	
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(4)
960.26/291.75	Complex Obligation (BEST)
960.26/291.75	
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(5)
960.26/291.75	Obligation:
960.26/291.75	Proved the lower bound n^1 for the following obligation:
960.26/291.75	
960.26/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
960.26/291.75	
960.26/291.75	
960.26/291.75	The TRS R consists of the following rules:
960.26/291.75	
960.26/291.75	   f(x, a(b(y))) -> f(a(b(b(x))), y)
960.26/291.75	   f(a(x), y) -> f(x, a(y))
960.26/291.75	   f(b(x), y) -> f(x, b(y))
960.26/291.75	
960.26/291.75	S is empty.
960.26/291.75	Rewrite Strategy: FULL
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(6) LowerBoundPropagationProof (FINISHED)
960.26/291.75	Propagated lower bound.
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(7)
960.26/291.75	BOUNDS(n^1, INF)
960.26/291.75	
960.26/291.75	----------------------------------------
960.26/291.75	
960.26/291.75	(8)
960.26/291.75	Obligation:
960.26/291.75	Analyzing the following TRS for decreasing loops:
960.26/291.75	
960.26/291.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
960.26/291.75	
960.26/291.75	
960.26/291.75	The TRS R consists of the following rules:
960.26/291.75	
960.26/291.75	   f(x, a(b(y))) -> f(a(b(b(x))), y)
960.26/291.75	   f(a(x), y) -> f(x, a(y))
960.26/291.75	   f(b(x), y) -> f(x, b(y))
960.26/291.75	
960.26/291.75	S is empty.
960.26/291.75	Rewrite Strategy: FULL
960.32/291.81	EOF