1106.98/291.51	WORST_CASE(Omega(n^1), ?)
1107.20/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1107.20/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1107.20/291.54	
1107.20/291.54	(0) CpxTRS
1107.20/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1107.20/291.54	(2) TRS for Loop Detection
1107.20/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1107.20/291.54	(4) BEST
1107.20/291.54	    (5) proven lower bound
1107.20/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1107.20/291.54	        (7) BOUNDS(n^1, INF)
1107.20/291.54	    (8) TRS for Loop Detection
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(0)
1107.20/291.54	Obligation:
1107.20/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	The TRS R consists of the following rules:
1107.20/291.54	
1107.20/291.54	   plus(x, y) -> plusIter(x, y, 0)
1107.20/291.54	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
1107.20/291.54	   ifPlus(true, x, y, z) -> y
1107.20/291.54	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
1107.20/291.54	   le(s(x), 0) -> false
1107.20/291.54	   le(0, y) -> true
1107.20/291.54	   le(s(x), s(y)) -> le(x, y)
1107.20/291.54	   sum(xs) -> sumIter(xs, 0)
1107.20/291.54	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
1107.20/291.54	   ifSum(true, xs, x, y) -> x
1107.20/291.54	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
1107.20/291.54	   isempty(nil) -> true
1107.20/291.54	   isempty(cons(x, xs)) -> false
1107.20/291.54	   head(nil) -> error
1107.20/291.54	   head(cons(x, xs)) -> x
1107.20/291.54	   tail(nil) -> nil
1107.20/291.54	   tail(cons(x, xs)) -> xs
1107.20/291.54	   a -> b
1107.20/291.54	   a -> c
1107.20/291.54	
1107.20/291.54	S is empty.
1107.20/291.54	Rewrite Strategy: INNERMOST
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1107.20/291.54	Transformed a relative TRS into a decreasing-loop problem.
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(2)
1107.20/291.54	Obligation:
1107.20/291.54	Analyzing the following TRS for decreasing loops:
1107.20/291.54	
1107.20/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	The TRS R consists of the following rules:
1107.20/291.54	
1107.20/291.54	   plus(x, y) -> plusIter(x, y, 0)
1107.20/291.54	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
1107.20/291.54	   ifPlus(true, x, y, z) -> y
1107.20/291.54	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
1107.20/291.54	   le(s(x), 0) -> false
1107.20/291.54	   le(0, y) -> true
1107.20/291.54	   le(s(x), s(y)) -> le(x, y)
1107.20/291.54	   sum(xs) -> sumIter(xs, 0)
1107.20/291.54	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
1107.20/291.54	   ifSum(true, xs, x, y) -> x
1107.20/291.54	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
1107.20/291.54	   isempty(nil) -> true
1107.20/291.54	   isempty(cons(x, xs)) -> false
1107.20/291.54	   head(nil) -> error
1107.20/291.54	   head(cons(x, xs)) -> x
1107.20/291.54	   tail(nil) -> nil
1107.20/291.54	   tail(cons(x, xs)) -> xs
1107.20/291.54	   a -> b
1107.20/291.54	   a -> c
1107.20/291.54	
1107.20/291.54	S is empty.
1107.20/291.54	Rewrite Strategy: INNERMOST
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
1107.20/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1107.20/291.54	
1107.20/291.54	The rewrite sequence
1107.20/291.54	
1107.20/291.54	le(s(x), s(y)) ->^+ le(x, y)
1107.20/291.54	
1107.20/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1107.20/291.54	
1107.20/291.54	The pumping substitution is [x / s(x), y / s(y)].
1107.20/291.54	
1107.20/291.54	The result substitution is [ ].
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(4)
1107.20/291.54	Complex Obligation (BEST)
1107.20/291.54	
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(5)
1107.20/291.54	Obligation:
1107.20/291.54	Proved the lower bound n^1 for the following obligation:
1107.20/291.54	
1107.20/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	The TRS R consists of the following rules:
1107.20/291.54	
1107.20/291.54	   plus(x, y) -> plusIter(x, y, 0)
1107.20/291.54	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
1107.20/291.54	   ifPlus(true, x, y, z) -> y
1107.20/291.54	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
1107.20/291.54	   le(s(x), 0) -> false
1107.20/291.54	   le(0, y) -> true
1107.20/291.54	   le(s(x), s(y)) -> le(x, y)
1107.20/291.54	   sum(xs) -> sumIter(xs, 0)
1107.20/291.54	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
1107.20/291.54	   ifSum(true, xs, x, y) -> x
1107.20/291.54	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
1107.20/291.54	   isempty(nil) -> true
1107.20/291.54	   isempty(cons(x, xs)) -> false
1107.20/291.54	   head(nil) -> error
1107.20/291.54	   head(cons(x, xs)) -> x
1107.20/291.54	   tail(nil) -> nil
1107.20/291.54	   tail(cons(x, xs)) -> xs
1107.20/291.54	   a -> b
1107.20/291.54	   a -> c
1107.20/291.54	
1107.20/291.54	S is empty.
1107.20/291.54	Rewrite Strategy: INNERMOST
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(6) LowerBoundPropagationProof (FINISHED)
1107.20/291.54	Propagated lower bound.
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(7)
1107.20/291.54	BOUNDS(n^1, INF)
1107.20/291.54	
1107.20/291.54	----------------------------------------
1107.20/291.54	
1107.20/291.54	(8)
1107.20/291.54	Obligation:
1107.20/291.54	Analyzing the following TRS for decreasing loops:
1107.20/291.54	
1107.20/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1107.20/291.54	
1107.20/291.54	
1107.20/291.54	The TRS R consists of the following rules:
1107.20/291.54	
1107.20/291.54	   plus(x, y) -> plusIter(x, y, 0)
1107.20/291.54	   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
1107.20/291.54	   ifPlus(true, x, y, z) -> y
1107.20/291.54	   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
1107.20/291.54	   le(s(x), 0) -> false
1107.20/291.54	   le(0, y) -> true
1107.20/291.54	   le(s(x), s(y)) -> le(x, y)
1107.20/291.54	   sum(xs) -> sumIter(xs, 0)
1107.20/291.54	   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
1107.20/291.54	   ifSum(true, xs, x, y) -> x
1107.20/291.54	   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
1107.20/291.54	   isempty(nil) -> true
1107.20/291.54	   isempty(cons(x, xs)) -> false
1107.20/291.54	   head(nil) -> error
1107.20/291.54	   head(cons(x, xs)) -> x
1107.20/291.54	   tail(nil) -> nil
1107.20/291.54	   tail(cons(x, xs)) -> xs
1107.20/291.54	   a -> b
1107.20/291.54	   a -> c
1107.20/291.54	
1107.20/291.54	S is empty.
1107.20/291.54	Rewrite Strategy: INNERMOST
1107.24/291.60	EOF