1088.09/292.01	WORST_CASE(Omega(n^1), ?)
1088.09/292.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1088.09/292.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1088.09/292.05	
1088.09/292.05	(0) CpxTRS
1088.09/292.05	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1088.09/292.05	(2) TRS for Loop Detection
1088.09/292.05	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1088.09/292.05	(4) BEST
1088.09/292.05	    (5) proven lower bound
1088.09/292.05	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1088.09/292.05	        (7) BOUNDS(n^1, INF)
1088.09/292.05	    (8) TRS for Loop Detection
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(0)
1088.09/292.05	Obligation:
1088.09/292.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	The TRS R consists of the following rules:
1088.09/292.05	
1088.09/292.05	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1088.09/292.05	   f(a(x), y) -> f(x, a(y))
1088.09/292.05	   f(b(x), y) -> f(x, b(y))
1088.09/292.05	
1088.09/292.05	S is empty.
1088.09/292.05	Rewrite Strategy: INNERMOST
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1088.09/292.05	Transformed a relative TRS into a decreasing-loop problem.
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(2)
1088.09/292.05	Obligation:
1088.09/292.05	Analyzing the following TRS for decreasing loops:
1088.09/292.05	
1088.09/292.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	The TRS R consists of the following rules:
1088.09/292.05	
1088.09/292.05	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1088.09/292.05	   f(a(x), y) -> f(x, a(y))
1088.09/292.05	   f(b(x), y) -> f(x, b(y))
1088.09/292.05	
1088.09/292.05	S is empty.
1088.09/292.05	Rewrite Strategy: INNERMOST
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(3) DecreasingLoopProof (LOWER BOUND(ID))
1088.09/292.05	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1088.09/292.05	
1088.09/292.05	The rewrite sequence
1088.09/292.05	
1088.09/292.05	f(x, a(b(y))) ->^+ a(f(a(b(x)), y))
1088.09/292.05	
1088.09/292.05	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1088.09/292.05	
1088.09/292.05	The pumping substitution is [y / a(b(y))].
1088.09/292.05	
1088.09/292.05	The result substitution is [x / a(b(x))].
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(4)
1088.09/292.05	Complex Obligation (BEST)
1088.09/292.05	
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(5)
1088.09/292.05	Obligation:
1088.09/292.05	Proved the lower bound n^1 for the following obligation:
1088.09/292.05	
1088.09/292.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	The TRS R consists of the following rules:
1088.09/292.05	
1088.09/292.05	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1088.09/292.05	   f(a(x), y) -> f(x, a(y))
1088.09/292.05	   f(b(x), y) -> f(x, b(y))
1088.09/292.05	
1088.09/292.05	S is empty.
1088.09/292.05	Rewrite Strategy: INNERMOST
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(6) LowerBoundPropagationProof (FINISHED)
1088.09/292.05	Propagated lower bound.
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(7)
1088.09/292.05	BOUNDS(n^1, INF)
1088.09/292.05	
1088.09/292.05	----------------------------------------
1088.09/292.05	
1088.09/292.05	(8)
1088.09/292.05	Obligation:
1088.09/292.05	Analyzing the following TRS for decreasing loops:
1088.09/292.05	
1088.09/292.05	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1088.09/292.05	
1088.09/292.05	
1088.09/292.05	The TRS R consists of the following rules:
1088.09/292.05	
1088.09/292.05	   f(x, a(b(y))) -> a(f(a(b(x)), y))
1088.09/292.05	   f(a(x), y) -> f(x, a(y))
1088.09/292.05	   f(b(x), y) -> f(x, b(y))
1088.09/292.05	
1088.09/292.05	S is empty.
1088.09/292.05	Rewrite Strategy: INNERMOST
1088.36/292.12	EOF