991.50/291.63	WORST_CASE(Omega(n^1), ?)
991.50/291.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
991.50/291.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
991.50/291.63	
991.50/291.63	
991.50/291.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
991.50/291.63	
991.50/291.63	(0) CpxTRS
991.50/291.63	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
991.50/291.63	(2) TRS for Loop Detection
991.50/291.63	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
991.50/291.63	(4) BEST
991.50/291.63	    (5) proven lower bound
991.50/291.63	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
991.50/291.63	        (7) BOUNDS(n^1, INF)
991.50/291.63	    (8) TRS for Loop Detection
991.50/291.63	
991.50/291.63	
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(0)
991.50/291.63	Obligation:
991.50/291.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
991.50/291.63	
991.50/291.63	
991.50/291.63	The TRS R consists of the following rules:
991.50/291.63	
991.50/291.63	   app(nil, k) -> k
991.50/291.63	   app(l, nil) -> l
991.50/291.63	   app(cons(x, l), k) -> cons(x, app(l, k))
991.50/291.63	   sum(cons(x, nil)) -> cons(x, nil)
991.50/291.63	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
991.50/291.63	   a(h, h, x) -> s(x)
991.50/291.63	   a(x, s(y), h) -> a(x, y, s(h))
991.50/291.63	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
991.50/291.63	   a(s(x), h, z) -> a(x, z, z)
991.50/291.63	
991.50/291.63	S is empty.
991.50/291.63	Rewrite Strategy: INNERMOST
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
991.50/291.63	Transformed a relative TRS into a decreasing-loop problem.
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(2)
991.50/291.63	Obligation:
991.50/291.63	Analyzing the following TRS for decreasing loops:
991.50/291.63	
991.50/291.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
991.50/291.63	
991.50/291.63	
991.50/291.63	The TRS R consists of the following rules:
991.50/291.63	
991.50/291.63	   app(nil, k) -> k
991.50/291.63	   app(l, nil) -> l
991.50/291.63	   app(cons(x, l), k) -> cons(x, app(l, k))
991.50/291.63	   sum(cons(x, nil)) -> cons(x, nil)
991.50/291.63	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
991.50/291.63	   a(h, h, x) -> s(x)
991.50/291.63	   a(x, s(y), h) -> a(x, y, s(h))
991.50/291.63	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
991.50/291.63	   a(s(x), h, z) -> a(x, z, z)
991.50/291.63	
991.50/291.63	S is empty.
991.50/291.63	Rewrite Strategy: INNERMOST
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(3) DecreasingLoopProof (LOWER BOUND(ID))
991.50/291.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
991.50/291.63	
991.50/291.63	The rewrite sequence
991.50/291.63	
991.50/291.63	app(cons(x, l), k) ->^+ cons(x, app(l, k))
991.50/291.63	
991.50/291.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
991.50/291.63	
991.50/291.63	The pumping substitution is [l / cons(x, l)].
991.50/291.63	
991.50/291.63	The result substitution is [ ].
991.50/291.63	
991.50/291.63	
991.50/291.63	
991.50/291.63	
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(4)
991.50/291.63	Complex Obligation (BEST)
991.50/291.63	
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(5)
991.50/291.63	Obligation:
991.50/291.63	Proved the lower bound n^1 for the following obligation:
991.50/291.63	
991.50/291.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
991.50/291.63	
991.50/291.63	
991.50/291.63	The TRS R consists of the following rules:
991.50/291.63	
991.50/291.63	   app(nil, k) -> k
991.50/291.63	   app(l, nil) -> l
991.50/291.63	   app(cons(x, l), k) -> cons(x, app(l, k))
991.50/291.63	   sum(cons(x, nil)) -> cons(x, nil)
991.50/291.63	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
991.50/291.63	   a(h, h, x) -> s(x)
991.50/291.63	   a(x, s(y), h) -> a(x, y, s(h))
991.50/291.63	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
991.50/291.63	   a(s(x), h, z) -> a(x, z, z)
991.50/291.63	
991.50/291.63	S is empty.
991.50/291.63	Rewrite Strategy: INNERMOST
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(6) LowerBoundPropagationProof (FINISHED)
991.50/291.63	Propagated lower bound.
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(7)
991.50/291.63	BOUNDS(n^1, INF)
991.50/291.63	
991.50/291.63	----------------------------------------
991.50/291.63	
991.50/291.63	(8)
991.50/291.63	Obligation:
991.50/291.63	Analyzing the following TRS for decreasing loops:
991.50/291.63	
991.50/291.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
991.50/291.63	
991.50/291.63	
991.50/291.63	The TRS R consists of the following rules:
991.50/291.63	
991.50/291.63	   app(nil, k) -> k
991.50/291.63	   app(l, nil) -> l
991.50/291.63	   app(cons(x, l), k) -> cons(x, app(l, k))
991.50/291.63	   sum(cons(x, nil)) -> cons(x, nil)
991.50/291.63	   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
991.50/291.63	   a(h, h, x) -> s(x)
991.50/291.63	   a(x, s(y), h) -> a(x, y, s(h))
991.50/291.63	   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
991.50/291.63	   a(s(x), h, z) -> a(x, z, z)
991.50/291.63	
991.50/291.63	S is empty.
991.50/291.63	Rewrite Strategy: INNERMOST
991.76/291.71	EOF