1083.45/291.51	WORST_CASE(Omega(n^1), ?)
1083.45/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1083.45/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1083.45/291.53	
1083.45/291.53	(0) CpxTRS
1083.45/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1083.45/291.53	(2) TRS for Loop Detection
1083.45/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1083.45/291.53	(4) BEST
1083.45/291.53	    (5) proven lower bound
1083.45/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1083.45/291.53	        (7) BOUNDS(n^1, INF)
1083.45/291.53	    (8) TRS for Loop Detection
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(0)
1083.45/291.53	Obligation:
1083.45/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	The TRS R consists of the following rules:
1083.45/291.53	
1083.45/291.53	   f(x, a(b(y))) -> f(a(a(x)), y)
1083.45/291.53	   f(x, b(a(y))) -> f(b(b(x)), y)
1083.45/291.53	   f(a(x), y) -> f(x, a(y))
1083.45/291.53	   f(b(x), y) -> f(x, b(y))
1083.45/291.53	
1083.45/291.53	S is empty.
1083.45/291.53	Rewrite Strategy: FULL
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1083.45/291.53	Transformed a relative TRS into a decreasing-loop problem.
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(2)
1083.45/291.53	Obligation:
1083.45/291.53	Analyzing the following TRS for decreasing loops:
1083.45/291.53	
1083.45/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	The TRS R consists of the following rules:
1083.45/291.53	
1083.45/291.53	   f(x, a(b(y))) -> f(a(a(x)), y)
1083.45/291.53	   f(x, b(a(y))) -> f(b(b(x)), y)
1083.45/291.53	   f(a(x), y) -> f(x, a(y))
1083.45/291.53	   f(b(x), y) -> f(x, b(y))
1083.45/291.53	
1083.45/291.53	S is empty.
1083.45/291.53	Rewrite Strategy: FULL
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
1083.45/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1083.45/291.53	
1083.45/291.53	The rewrite sequence
1083.45/291.53	
1083.45/291.53	f(x, a(b(y))) ->^+ f(a(a(x)), y)
1083.45/291.53	
1083.45/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1083.45/291.53	
1083.45/291.53	The pumping substitution is [y / a(b(y))].
1083.45/291.53	
1083.45/291.53	The result substitution is [x / a(a(x))].
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(4)
1083.45/291.53	Complex Obligation (BEST)
1083.45/291.53	
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(5)
1083.45/291.53	Obligation:
1083.45/291.53	Proved the lower bound n^1 for the following obligation:
1083.45/291.53	
1083.45/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	The TRS R consists of the following rules:
1083.45/291.53	
1083.45/291.53	   f(x, a(b(y))) -> f(a(a(x)), y)
1083.45/291.53	   f(x, b(a(y))) -> f(b(b(x)), y)
1083.45/291.53	   f(a(x), y) -> f(x, a(y))
1083.45/291.53	   f(b(x), y) -> f(x, b(y))
1083.45/291.53	
1083.45/291.53	S is empty.
1083.45/291.53	Rewrite Strategy: FULL
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(6) LowerBoundPropagationProof (FINISHED)
1083.45/291.53	Propagated lower bound.
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(7)
1083.45/291.53	BOUNDS(n^1, INF)
1083.45/291.53	
1083.45/291.53	----------------------------------------
1083.45/291.53	
1083.45/291.53	(8)
1083.45/291.53	Obligation:
1083.45/291.53	Analyzing the following TRS for decreasing loops:
1083.45/291.53	
1083.45/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1083.45/291.53	
1083.45/291.53	
1083.45/291.53	The TRS R consists of the following rules:
1083.45/291.53	
1083.45/291.53	   f(x, a(b(y))) -> f(a(a(x)), y)
1083.45/291.53	   f(x, b(a(y))) -> f(b(b(x)), y)
1083.45/291.53	   f(a(x), y) -> f(x, a(y))
1083.45/291.53	   f(b(x), y) -> f(x, b(y))
1083.45/291.53	
1083.45/291.53	S is empty.
1083.45/291.53	Rewrite Strategy: FULL
1083.77/291.60	EOF