1102.91/291.51	WORST_CASE(Omega(n^1), ?)
1105.30/292.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1105.30/292.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.30/292.12	
1105.30/292.12	(0) CpxTRS
1105.30/292.12	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1105.30/292.12	(2) TRS for Loop Detection
1105.30/292.12	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1105.30/292.12	(4) BEST
1105.30/292.12	    (5) proven lower bound
1105.30/292.12	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1105.30/292.12	        (7) BOUNDS(n^1, INF)
1105.30/292.12	    (8) TRS for Loop Detection
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(0)
1105.30/292.12	Obligation:
1105.30/292.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	The TRS R consists of the following rules:
1105.30/292.12	
1105.30/292.12	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1105.30/292.12	   f(a(x), y) -> f(x, a(y))
1105.30/292.12	   f(b(x), y) -> f(x, b(y))
1105.30/292.12	
1105.30/292.12	S is empty.
1105.30/292.12	Rewrite Strategy: FULL
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1105.30/292.12	Transformed a relative TRS into a decreasing-loop problem.
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(2)
1105.30/292.12	Obligation:
1105.30/292.12	Analyzing the following TRS for decreasing loops:
1105.30/292.12	
1105.30/292.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	The TRS R consists of the following rules:
1105.30/292.12	
1105.30/292.12	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1105.30/292.12	   f(a(x), y) -> f(x, a(y))
1105.30/292.12	   f(b(x), y) -> f(x, b(y))
1105.30/292.12	
1105.30/292.12	S is empty.
1105.30/292.12	Rewrite Strategy: FULL
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(3) DecreasingLoopProof (LOWER BOUND(ID))
1105.30/292.12	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1105.30/292.12	
1105.30/292.12	The rewrite sequence
1105.30/292.12	
1105.30/292.12	f(x, a(b(y))) ->^+ a(f(a(b(x)), y))
1105.30/292.12	
1105.30/292.12	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1105.30/292.12	
1105.30/292.12	The pumping substitution is [y / a(b(y))].
1105.30/292.12	
1105.30/292.12	The result substitution is [x / a(b(x))].
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(4)
1105.30/292.12	Complex Obligation (BEST)
1105.30/292.12	
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(5)
1105.30/292.12	Obligation:
1105.30/292.12	Proved the lower bound n^1 for the following obligation:
1105.30/292.12	
1105.30/292.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	The TRS R consists of the following rules:
1105.30/292.12	
1105.30/292.12	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1105.30/292.12	   f(a(x), y) -> f(x, a(y))
1105.30/292.12	   f(b(x), y) -> f(x, b(y))
1105.30/292.12	
1105.30/292.12	S is empty.
1105.30/292.12	Rewrite Strategy: FULL
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(6) LowerBoundPropagationProof (FINISHED)
1105.30/292.12	Propagated lower bound.
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(7)
1105.30/292.12	BOUNDS(n^1, INF)
1105.30/292.12	
1105.30/292.12	----------------------------------------
1105.30/292.12	
1105.30/292.12	(8)
1105.30/292.12	Obligation:
1105.30/292.12	Analyzing the following TRS for decreasing loops:
1105.30/292.12	
1105.30/292.12	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1105.30/292.12	
1105.30/292.12	
1105.30/292.12	The TRS R consists of the following rules:
1105.30/292.12	
1105.30/292.12	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1105.30/292.12	   f(a(x), y) -> f(x, a(y))
1105.30/292.12	   f(b(x), y) -> f(x, b(y))
1105.30/292.12	
1105.30/292.12	S is empty.
1105.30/292.12	Rewrite Strategy: FULL
1105.52/292.19	EOF