318.26/291.52	WORST_CASE(Omega(n^1), ?)
318.26/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
318.26/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
318.26/291.52	
318.26/291.52	
318.26/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.26/291.52	
318.26/291.52	(0) CpxTRS
318.26/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
318.26/291.52	(2) TRS for Loop Detection
318.26/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
318.26/291.52	(4) BEST
318.26/291.52	    (5) proven lower bound
318.26/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
318.26/291.52	        (7) BOUNDS(n^1, INF)
318.26/291.52	    (8) TRS for Loop Detection
318.26/291.52	
318.26/291.52	
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(0)
318.26/291.52	Obligation:
318.26/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.26/291.52	
318.26/291.52	
318.26/291.52	The TRS R consists of the following rules:
318.26/291.52	
318.26/291.52	   +(0, y) -> y
318.26/291.52	   +(s(x), y) -> s(+(x, y))
318.26/291.52	   ++(nil, ys) -> ys
318.26/291.52	   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
318.26/291.52	   sum(:(x, nil)) -> :(x, nil)
318.26/291.52	   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
318.26/291.52	   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
318.26/291.52	   -(x, 0) -> x
318.26/291.52	   -(0, s(y)) -> 0
318.26/291.52	   -(s(x), s(y)) -> -(x, y)
318.26/291.52	   quot(0, s(y)) -> 0
318.26/291.52	   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
318.26/291.52	   length(nil) -> 0
318.26/291.52	   length(:(x, xs)) -> s(length(xs))
318.26/291.52	   hd(:(x, xs)) -> x
318.26/291.52	   avg(xs) -> quot(hd(sum(xs)), length(xs))
318.26/291.52	
318.26/291.52	S is empty.
318.26/291.52	Rewrite Strategy: FULL
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
318.26/291.52	Transformed a relative TRS into a decreasing-loop problem.
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(2)
318.26/291.52	Obligation:
318.26/291.52	Analyzing the following TRS for decreasing loops:
318.26/291.52	
318.26/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.26/291.52	
318.26/291.52	
318.26/291.52	The TRS R consists of the following rules:
318.26/291.52	
318.26/291.52	   +(0, y) -> y
318.26/291.52	   +(s(x), y) -> s(+(x, y))
318.26/291.52	   ++(nil, ys) -> ys
318.26/291.52	   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
318.26/291.52	   sum(:(x, nil)) -> :(x, nil)
318.26/291.52	   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
318.26/291.52	   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
318.26/291.52	   -(x, 0) -> x
318.26/291.52	   -(0, s(y)) -> 0
318.26/291.52	   -(s(x), s(y)) -> -(x, y)
318.26/291.52	   quot(0, s(y)) -> 0
318.26/291.52	   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
318.26/291.52	   length(nil) -> 0
318.26/291.52	   length(:(x, xs)) -> s(length(xs))
318.26/291.52	   hd(:(x, xs)) -> x
318.26/291.52	   avg(xs) -> quot(hd(sum(xs)), length(xs))
318.26/291.52	
318.26/291.52	S is empty.
318.26/291.52	Rewrite Strategy: FULL
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
318.26/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
318.26/291.52	
318.26/291.52	The rewrite sequence
318.26/291.52	
318.26/291.52	-(s(x), s(y)) ->^+ -(x, y)
318.26/291.52	
318.26/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
318.26/291.52	
318.26/291.52	The pumping substitution is [x / s(x), y / s(y)].
318.26/291.52	
318.26/291.52	The result substitution is [ ].
318.26/291.52	
318.26/291.52	
318.26/291.52	
318.26/291.52	
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(4)
318.26/291.52	Complex Obligation (BEST)
318.26/291.52	
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(5)
318.26/291.52	Obligation:
318.26/291.52	Proved the lower bound n^1 for the following obligation:
318.26/291.52	
318.26/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.26/291.52	
318.26/291.52	
318.26/291.52	The TRS R consists of the following rules:
318.26/291.52	
318.26/291.52	   +(0, y) -> y
318.26/291.52	   +(s(x), y) -> s(+(x, y))
318.26/291.52	   ++(nil, ys) -> ys
318.26/291.52	   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
318.26/291.52	   sum(:(x, nil)) -> :(x, nil)
318.26/291.52	   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
318.26/291.52	   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
318.26/291.52	   -(x, 0) -> x
318.26/291.52	   -(0, s(y)) -> 0
318.26/291.52	   -(s(x), s(y)) -> -(x, y)
318.26/291.52	   quot(0, s(y)) -> 0
318.26/291.52	   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
318.26/291.52	   length(nil) -> 0
318.26/291.52	   length(:(x, xs)) -> s(length(xs))
318.26/291.52	   hd(:(x, xs)) -> x
318.26/291.52	   avg(xs) -> quot(hd(sum(xs)), length(xs))
318.26/291.52	
318.26/291.52	S is empty.
318.26/291.52	Rewrite Strategy: FULL
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(6) LowerBoundPropagationProof (FINISHED)
318.26/291.52	Propagated lower bound.
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(7)
318.26/291.52	BOUNDS(n^1, INF)
318.26/291.52	
318.26/291.52	----------------------------------------
318.26/291.52	
318.26/291.52	(8)
318.26/291.52	Obligation:
318.26/291.52	Analyzing the following TRS for decreasing loops:
318.26/291.52	
318.26/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.26/291.52	
318.26/291.52	
318.26/291.52	The TRS R consists of the following rules:
318.26/291.52	
318.26/291.52	   +(0, y) -> y
318.26/291.52	   +(s(x), y) -> s(+(x, y))
318.26/291.52	   ++(nil, ys) -> ys
318.26/291.52	   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
318.26/291.52	   sum(:(x, nil)) -> :(x, nil)
318.26/291.52	   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
318.26/291.52	   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
318.26/291.52	   -(x, 0) -> x
318.26/291.52	   -(0, s(y)) -> 0
318.26/291.52	   -(s(x), s(y)) -> -(x, y)
318.26/291.52	   quot(0, s(y)) -> 0
318.26/291.52	   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
318.26/291.52	   length(nil) -> 0
318.26/291.52	   length(:(x, xs)) -> s(length(xs))
318.26/291.52	   hd(:(x, xs)) -> x
318.26/291.52	   avg(xs) -> quot(hd(sum(xs)), length(xs))
318.26/291.52	
318.26/291.52	S is empty.
318.26/291.52	Rewrite Strategy: FULL
318.26/291.56	EOF