310.48/291.49	WORST_CASE(Omega(n^1), ?)
310.48/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
310.48/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
310.48/291.50	
310.48/291.50	
310.48/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.48/291.50	
310.48/291.50	(0) CpxTRS
310.48/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
310.48/291.50	(2) TRS for Loop Detection
310.48/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
310.48/291.50	(4) BEST
310.48/291.50	    (5) proven lower bound
310.48/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
310.48/291.50	        (7) BOUNDS(n^1, INF)
310.48/291.50	    (8) TRS for Loop Detection
310.48/291.50	
310.48/291.50	
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(0)
310.48/291.50	Obligation:
310.48/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.48/291.50	
310.48/291.50	
310.48/291.50	The TRS R consists of the following rules:
310.48/291.50	
310.48/291.50	   plus(x, y) -> plusIter(x, y, 0)
310.48/291.50	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
310.48/291.50	   ifPlus(true, x, y, z) -> y
310.48/291.50	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
310.48/291.50	   le(s(x), 0) -> false
310.48/291.50	   le(0, y) -> true
310.48/291.50	   le(s(x), s(y)) -> le(x, y)
310.48/291.50	   sum(xs) -> sumIter(xs, 0)
310.48/291.50	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
310.48/291.50	   ifSum(true, xs, x, y) -> x
310.48/291.50	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
310.48/291.50	   isempty(nil) -> true
310.48/291.50	   isempty(cons(x, xs)) -> false
310.48/291.50	   head(nil) -> error
310.48/291.50	   head(cons(x, xs)) -> x
310.48/291.50	   tail(nil) -> nil
310.48/291.50	   tail(cons(x, xs)) -> xs
310.48/291.50	   a -> b
310.48/291.50	   a -> c
310.48/291.50	
310.48/291.50	S is empty.
310.48/291.50	Rewrite Strategy: FULL
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
310.48/291.50	Transformed a relative TRS into a decreasing-loop problem.
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(2)
310.48/291.50	Obligation:
310.48/291.50	Analyzing the following TRS for decreasing loops:
310.48/291.50	
310.48/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.48/291.50	
310.48/291.50	
310.48/291.50	The TRS R consists of the following rules:
310.48/291.50	
310.48/291.50	   plus(x, y) -> plusIter(x, y, 0)
310.48/291.50	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
310.48/291.50	   ifPlus(true, x, y, z) -> y
310.48/291.50	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
310.48/291.50	   le(s(x), 0) -> false
310.48/291.50	   le(0, y) -> true
310.48/291.50	   le(s(x), s(y)) -> le(x, y)
310.48/291.50	   sum(xs) -> sumIter(xs, 0)
310.48/291.50	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
310.48/291.50	   ifSum(true, xs, x, y) -> x
310.48/291.50	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
310.48/291.50	   isempty(nil) -> true
310.48/291.50	   isempty(cons(x, xs)) -> false
310.48/291.50	   head(nil) -> error
310.48/291.50	   head(cons(x, xs)) -> x
310.48/291.50	   tail(nil) -> nil
310.48/291.50	   tail(cons(x, xs)) -> xs
310.48/291.50	   a -> b
310.48/291.50	   a -> c
310.48/291.50	
310.48/291.50	S is empty.
310.48/291.50	Rewrite Strategy: FULL
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
310.48/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
310.48/291.50	
310.48/291.50	The rewrite sequence
310.48/291.50	
310.48/291.50	le(s(x), s(y)) ->^+ le(x, y)
310.48/291.50	
310.48/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
310.48/291.50	
310.48/291.50	The pumping substitution is [x / s(x), y / s(y)].
310.48/291.50	
310.48/291.50	The result substitution is [ ].
310.48/291.50	
310.48/291.50	
310.48/291.50	
310.48/291.50	
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(4)
310.48/291.50	Complex Obligation (BEST)
310.48/291.50	
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(5)
310.48/291.50	Obligation:
310.48/291.50	Proved the lower bound n^1 for the following obligation:
310.48/291.50	
310.48/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.48/291.50	
310.48/291.50	
310.48/291.50	The TRS R consists of the following rules:
310.48/291.50	
310.48/291.50	   plus(x, y) -> plusIter(x, y, 0)
310.48/291.50	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
310.48/291.50	   ifPlus(true, x, y, z) -> y
310.48/291.50	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
310.48/291.50	   le(s(x), 0) -> false
310.48/291.50	   le(0, y) -> true
310.48/291.50	   le(s(x), s(y)) -> le(x, y)
310.48/291.50	   sum(xs) -> sumIter(xs, 0)
310.48/291.50	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
310.48/291.50	   ifSum(true, xs, x, y) -> x
310.48/291.50	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
310.48/291.50	   isempty(nil) -> true
310.48/291.50	   isempty(cons(x, xs)) -> false
310.48/291.50	   head(nil) -> error
310.48/291.50	   head(cons(x, xs)) -> x
310.48/291.50	   tail(nil) -> nil
310.48/291.50	   tail(cons(x, xs)) -> xs
310.48/291.50	   a -> b
310.48/291.50	   a -> c
310.48/291.50	
310.48/291.50	S is empty.
310.48/291.50	Rewrite Strategy: FULL
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(6) LowerBoundPropagationProof (FINISHED)
310.48/291.50	Propagated lower bound.
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(7)
310.48/291.50	BOUNDS(n^1, INF)
310.48/291.50	
310.48/291.50	----------------------------------------
310.48/291.50	
310.48/291.50	(8)
310.48/291.50	Obligation:
310.48/291.50	Analyzing the following TRS for decreasing loops:
310.48/291.50	
310.48/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.48/291.50	
310.48/291.50	
310.48/291.50	The TRS R consists of the following rules:
310.48/291.50	
310.48/291.50	   plus(x, y) -> plusIter(x, y, 0)
310.48/291.50	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
310.48/291.50	   ifPlus(true, x, y, z) -> y
310.48/291.50	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
310.48/291.50	   le(s(x), 0) -> false
310.48/291.50	   le(0, y) -> true
310.48/291.50	   le(s(x), s(y)) -> le(x, y)
310.48/291.50	   sum(xs) -> sumIter(xs, 0)
310.48/291.50	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
310.48/291.50	   ifSum(true, xs, x, y) -> x
310.48/291.50	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
310.48/291.50	   isempty(nil) -> true
310.48/291.50	   isempty(cons(x, xs)) -> false
310.48/291.50	   head(nil) -> error
310.48/291.50	   head(cons(x, xs)) -> x
310.48/291.50	   tail(nil) -> nil
310.48/291.50	   tail(cons(x, xs)) -> xs
310.48/291.50	   a -> b
310.48/291.50	   a -> c
310.48/291.50	
310.48/291.50	S is empty.
310.48/291.50	Rewrite Strategy: FULL
310.56/291.52	EOF