1121.79/291.66	WORST_CASE(Omega(n^1), ?)
1121.79/291.67	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1121.79/291.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.79/291.67	
1121.79/291.67	(0) CpxTRS
1121.79/291.67	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1121.79/291.67	(2) TRS for Loop Detection
1121.79/291.67	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1121.79/291.67	(4) BEST
1121.79/291.67	    (5) proven lower bound
1121.79/291.67	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1121.79/291.67	        (7) BOUNDS(n^1, INF)
1121.79/291.67	    (8) TRS for Loop Detection
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(0)
1121.79/291.67	Obligation:
1121.79/291.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	The TRS R consists of the following rules:
1121.79/291.67	
1121.79/291.67	   f(x, a(b(y))) -> f(a(b(x)), y)
1121.79/291.67	   f(x, b(c(y))) -> f(b(c(x)), y)
1121.79/291.67	   f(x, c(a(y))) -> f(c(a(x)), y)
1121.79/291.67	   f(a(x), y) -> f(x, a(y))
1121.79/291.67	   f(b(x), y) -> f(x, b(y))
1121.79/291.67	   f(c(x), y) -> f(x, c(y))
1121.79/291.67	
1121.79/291.67	S is empty.
1121.79/291.67	Rewrite Strategy: FULL
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1121.79/291.67	Transformed a relative TRS into a decreasing-loop problem.
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(2)
1121.79/291.67	Obligation:
1121.79/291.67	Analyzing the following TRS for decreasing loops:
1121.79/291.67	
1121.79/291.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	The TRS R consists of the following rules:
1121.79/291.67	
1121.79/291.67	   f(x, a(b(y))) -> f(a(b(x)), y)
1121.79/291.67	   f(x, b(c(y))) -> f(b(c(x)), y)
1121.79/291.67	   f(x, c(a(y))) -> f(c(a(x)), y)
1121.79/291.67	   f(a(x), y) -> f(x, a(y))
1121.79/291.67	   f(b(x), y) -> f(x, b(y))
1121.79/291.67	   f(c(x), y) -> f(x, c(y))
1121.79/291.67	
1121.79/291.67	S is empty.
1121.79/291.67	Rewrite Strategy: FULL
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(3) DecreasingLoopProof (LOWER BOUND(ID))
1121.79/291.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1121.79/291.67	
1121.79/291.67	The rewrite sequence
1121.79/291.67	
1121.79/291.67	f(c(x), y) ->^+ f(x, c(y))
1121.79/291.67	
1121.79/291.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1121.79/291.67	
1121.79/291.67	The pumping substitution is [x / c(x)].
1121.79/291.67	
1121.79/291.67	The result substitution is [y / c(y)].
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(4)
1121.79/291.67	Complex Obligation (BEST)
1121.79/291.67	
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(5)
1121.79/291.67	Obligation:
1121.79/291.67	Proved the lower bound n^1 for the following obligation:
1121.79/291.67	
1121.79/291.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	The TRS R consists of the following rules:
1121.79/291.67	
1121.79/291.67	   f(x, a(b(y))) -> f(a(b(x)), y)
1121.79/291.67	   f(x, b(c(y))) -> f(b(c(x)), y)
1121.79/291.67	   f(x, c(a(y))) -> f(c(a(x)), y)
1121.79/291.67	   f(a(x), y) -> f(x, a(y))
1121.79/291.67	   f(b(x), y) -> f(x, b(y))
1121.79/291.67	   f(c(x), y) -> f(x, c(y))
1121.79/291.67	
1121.79/291.67	S is empty.
1121.79/291.67	Rewrite Strategy: FULL
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(6) LowerBoundPropagationProof (FINISHED)
1121.79/291.67	Propagated lower bound.
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(7)
1121.79/291.67	BOUNDS(n^1, INF)
1121.79/291.67	
1121.79/291.67	----------------------------------------
1121.79/291.67	
1121.79/291.67	(8)
1121.79/291.67	Obligation:
1121.79/291.67	Analyzing the following TRS for decreasing loops:
1121.79/291.67	
1121.79/291.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.79/291.67	
1121.79/291.67	
1121.79/291.67	The TRS R consists of the following rules:
1121.79/291.67	
1121.79/291.67	   f(x, a(b(y))) -> f(a(b(x)), y)
1121.79/291.67	   f(x, b(c(y))) -> f(b(c(x)), y)
1121.79/291.67	   f(x, c(a(y))) -> f(c(a(x)), y)
1121.79/291.67	   f(a(x), y) -> f(x, a(y))
1121.79/291.67	   f(b(x), y) -> f(x, b(y))
1121.79/291.67	   f(c(x), y) -> f(x, c(y))
1121.79/291.67	
1121.79/291.67	S is empty.
1121.79/291.67	Rewrite Strategy: FULL
1122.00/291.76	EOF