1072.34/291.51	WORST_CASE(Omega(n^1), ?)
1072.80/291.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1072.80/291.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1072.80/291.65	
1072.80/291.65	(0) CpxTRS
1072.80/291.65	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1072.80/291.65	(2) TRS for Loop Detection
1072.80/291.65	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1072.80/291.65	(4) BEST
1072.80/291.65	    (5) proven lower bound
1072.80/291.65	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1072.80/291.65	        (7) BOUNDS(n^1, INF)
1072.80/291.65	    (8) TRS for Loop Detection
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(0)
1072.80/291.65	Obligation:
1072.80/291.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	The TRS R consists of the following rules:
1072.80/291.65	
1072.80/291.65	   f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))), y)
1072.80/291.65	   f(a(x), y) -> f(x, a(y))
1072.80/291.65	   f(b(x), y) -> f(x, b(y))
1072.80/291.65	
1072.80/291.65	S is empty.
1072.80/291.65	Rewrite Strategy: INNERMOST
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1072.80/291.65	Transformed a relative TRS into a decreasing-loop problem.
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(2)
1072.80/291.65	Obligation:
1072.80/291.65	Analyzing the following TRS for decreasing loops:
1072.80/291.65	
1072.80/291.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	The TRS R consists of the following rules:
1072.80/291.65	
1072.80/291.65	   f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))), y)
1072.80/291.65	   f(a(x), y) -> f(x, a(y))
1072.80/291.65	   f(b(x), y) -> f(x, b(y))
1072.80/291.65	
1072.80/291.65	S is empty.
1072.80/291.65	Rewrite Strategy: INNERMOST
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(3) DecreasingLoopProof (LOWER BOUND(ID))
1072.80/291.65	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1072.80/291.65	
1072.80/291.65	The rewrite sequence
1072.80/291.65	
1072.80/291.65	f(x, a(a(b(b(y))))) ->^+ f(a(a(a(b(b(b(x)))))), y)
1072.80/291.65	
1072.80/291.65	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1072.80/291.65	
1072.80/291.65	The pumping substitution is [y / a(a(b(b(y))))].
1072.80/291.65	
1072.80/291.65	The result substitution is [x / a(a(a(b(b(b(x))))))].
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(4)
1072.80/291.65	Complex Obligation (BEST)
1072.80/291.65	
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(5)
1072.80/291.65	Obligation:
1072.80/291.65	Proved the lower bound n^1 for the following obligation:
1072.80/291.65	
1072.80/291.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	The TRS R consists of the following rules:
1072.80/291.65	
1072.80/291.65	   f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))), y)
1072.80/291.65	   f(a(x), y) -> f(x, a(y))
1072.80/291.65	   f(b(x), y) -> f(x, b(y))
1072.80/291.65	
1072.80/291.65	S is empty.
1072.80/291.65	Rewrite Strategy: INNERMOST
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(6) LowerBoundPropagationProof (FINISHED)
1072.80/291.65	Propagated lower bound.
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(7)
1072.80/291.65	BOUNDS(n^1, INF)
1072.80/291.65	
1072.80/291.65	----------------------------------------
1072.80/291.65	
1072.80/291.65	(8)
1072.80/291.65	Obligation:
1072.80/291.65	Analyzing the following TRS for decreasing loops:
1072.80/291.65	
1072.80/291.65	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1072.80/291.65	
1072.80/291.65	
1072.80/291.65	The TRS R consists of the following rules:
1072.80/291.65	
1072.80/291.65	   f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))), y)
1072.80/291.65	   f(a(x), y) -> f(x, a(y))
1072.80/291.65	   f(b(x), y) -> f(x, b(y))
1072.80/291.65	
1072.80/291.65	S is empty.
1072.80/291.65	Rewrite Strategy: INNERMOST
1072.99/291.71	EOF