1022.53/291.50	WORST_CASE(Omega(n^1), ?)
1028.57/292.97	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1028.57/292.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1028.57/292.97	
1028.57/292.97	(0) CpxTRS
1028.57/292.97	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1028.57/292.97	(2) TRS for Loop Detection
1028.57/292.97	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1028.57/292.97	(4) BEST
1028.57/292.97	    (5) proven lower bound
1028.57/292.97	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1028.57/292.97	        (7) BOUNDS(n^1, INF)
1028.57/292.97	    (8) TRS for Loop Detection
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(0)
1028.57/292.97	Obligation:
1028.57/292.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	The TRS R consists of the following rules:
1028.57/292.97	
1028.57/292.97	   f(x, a(b(y))) -> f(a(a(b(x))), y)
1028.57/292.97	   f(a(x), y) -> f(x, a(y))
1028.57/292.97	   f(b(x), y) -> f(x, b(y))
1028.57/292.97	
1028.57/292.97	S is empty.
1028.57/292.97	Rewrite Strategy: FULL
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1028.57/292.97	Transformed a relative TRS into a decreasing-loop problem.
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(2)
1028.57/292.97	Obligation:
1028.57/292.97	Analyzing the following TRS for decreasing loops:
1028.57/292.97	
1028.57/292.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	The TRS R consists of the following rules:
1028.57/292.97	
1028.57/292.97	   f(x, a(b(y))) -> f(a(a(b(x))), y)
1028.57/292.97	   f(a(x), y) -> f(x, a(y))
1028.57/292.97	   f(b(x), y) -> f(x, b(y))
1028.57/292.97	
1028.57/292.97	S is empty.
1028.57/292.97	Rewrite Strategy: FULL
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(3) DecreasingLoopProof (LOWER BOUND(ID))
1028.57/292.97	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1028.57/292.97	
1028.57/292.97	The rewrite sequence
1028.57/292.97	
1028.57/292.97	f(b(x), y) ->^+ f(x, b(y))
1028.57/292.97	
1028.57/292.97	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1028.57/292.97	
1028.57/292.97	The pumping substitution is [x / b(x)].
1028.57/292.97	
1028.57/292.97	The result substitution is [y / b(y)].
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(4)
1028.57/292.97	Complex Obligation (BEST)
1028.57/292.97	
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(5)
1028.57/292.97	Obligation:
1028.57/292.97	Proved the lower bound n^1 for the following obligation:
1028.57/292.97	
1028.57/292.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	The TRS R consists of the following rules:
1028.57/292.97	
1028.57/292.97	   f(x, a(b(y))) -> f(a(a(b(x))), y)
1028.57/292.97	   f(a(x), y) -> f(x, a(y))
1028.57/292.97	   f(b(x), y) -> f(x, b(y))
1028.57/292.97	
1028.57/292.97	S is empty.
1028.57/292.97	Rewrite Strategy: FULL
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(6) LowerBoundPropagationProof (FINISHED)
1028.57/292.97	Propagated lower bound.
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(7)
1028.57/292.97	BOUNDS(n^1, INF)
1028.57/292.97	
1028.57/292.97	----------------------------------------
1028.57/292.97	
1028.57/292.97	(8)
1028.57/292.97	Obligation:
1028.57/292.97	Analyzing the following TRS for decreasing loops:
1028.57/292.97	
1028.57/292.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1028.57/292.97	
1028.57/292.97	
1028.57/292.97	The TRS R consists of the following rules:
1028.57/292.97	
1028.57/292.97	   f(x, a(b(y))) -> f(a(a(b(x))), y)
1028.57/292.97	   f(a(x), y) -> f(x, a(y))
1028.57/292.97	   f(b(x), y) -> f(x, b(y))
1028.57/292.97	
1028.57/292.97	S is empty.
1028.57/292.97	Rewrite Strategy: FULL
1028.68/293.03	EOF