951.95/291.76	WORST_CASE(Omega(n^1), ?)
951.95/291.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
951.95/291.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
951.95/291.77	
951.95/291.77	
951.95/291.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
951.95/291.77	
951.95/291.77	(0) CpxTRS
951.95/291.77	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
951.95/291.77	(2) TRS for Loop Detection
951.95/291.77	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
951.95/291.77	(4) BEST
951.95/291.77	    (5) proven lower bound
951.95/291.77	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
951.95/291.77	        (7) BOUNDS(n^1, INF)
951.95/291.77	    (8) TRS for Loop Detection
951.95/291.77	
951.95/291.77	
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(0)
951.95/291.77	Obligation:
951.95/291.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
951.95/291.77	
951.95/291.77	
951.95/291.77	The TRS R consists of the following rules:
951.95/291.77	
951.95/291.77	   f(x, a(b(y))) -> f(a(b(b(x))), y)
951.95/291.77	   f(a(x), y) -> f(x, a(y))
951.95/291.77	   f(b(x), y) -> f(x, b(y))
951.95/291.77	
951.95/291.77	S is empty.
951.95/291.77	Rewrite Strategy: INNERMOST
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
951.95/291.77	Transformed a relative TRS into a decreasing-loop problem.
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(2)
951.95/291.77	Obligation:
951.95/291.77	Analyzing the following TRS for decreasing loops:
951.95/291.77	
951.95/291.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
951.95/291.77	
951.95/291.77	
951.95/291.77	The TRS R consists of the following rules:
951.95/291.77	
951.95/291.77	   f(x, a(b(y))) -> f(a(b(b(x))), y)
951.95/291.77	   f(a(x), y) -> f(x, a(y))
951.95/291.77	   f(b(x), y) -> f(x, b(y))
951.95/291.77	
951.95/291.77	S is empty.
951.95/291.77	Rewrite Strategy: INNERMOST
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(3) DecreasingLoopProof (LOWER BOUND(ID))
951.95/291.77	The following loop(s) give(s) rise to the lower bound Omega(n^1):
951.95/291.77	
951.95/291.77	The rewrite sequence
951.95/291.77	
951.95/291.77	f(x, a(b(y))) ->^+ f(a(b(b(x))), y)
951.95/291.77	
951.95/291.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
951.95/291.77	
951.95/291.77	The pumping substitution is [y / a(b(y))].
951.95/291.77	
951.95/291.77	The result substitution is [x / a(b(b(x)))].
951.95/291.77	
951.95/291.77	
951.95/291.77	
951.95/291.77	
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(4)
951.95/291.77	Complex Obligation (BEST)
951.95/291.77	
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(5)
951.95/291.77	Obligation:
951.95/291.77	Proved the lower bound n^1 for the following obligation:
951.95/291.77	
951.95/291.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
951.95/291.77	
951.95/291.77	
951.95/291.77	The TRS R consists of the following rules:
951.95/291.77	
951.95/291.77	   f(x, a(b(y))) -> f(a(b(b(x))), y)
951.95/291.77	   f(a(x), y) -> f(x, a(y))
951.95/291.77	   f(b(x), y) -> f(x, b(y))
951.95/291.77	
951.95/291.77	S is empty.
951.95/291.77	Rewrite Strategy: INNERMOST
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(6) LowerBoundPropagationProof (FINISHED)
951.95/291.77	Propagated lower bound.
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(7)
951.95/291.77	BOUNDS(n^1, INF)
951.95/291.77	
951.95/291.77	----------------------------------------
951.95/291.77	
951.95/291.77	(8)
951.95/291.77	Obligation:
951.95/291.77	Analyzing the following TRS for decreasing loops:
951.95/291.77	
951.95/291.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
951.95/291.77	
951.95/291.77	
951.95/291.77	The TRS R consists of the following rules:
951.95/291.77	
951.95/291.77	   f(x, a(b(y))) -> f(a(b(b(x))), y)
951.95/291.77	   f(a(x), y) -> f(x, a(y))
951.95/291.77	   f(b(x), y) -> f(x, b(y))
951.95/291.77	
951.95/291.77	S is empty.
951.95/291.77	Rewrite Strategy: INNERMOST
952.18/291.84	EOF