311.23/291.53	WORST_CASE(Omega(n^1), ?)
311.23/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
311.23/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.23/291.54	
311.23/291.54	
311.23/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.23/291.54	
311.23/291.54	(0) CpxTRS
311.23/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
311.23/291.54	(2) TRS for Loop Detection
311.23/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
311.23/291.54	(4) BEST
311.23/291.54	    (5) proven lower bound
311.23/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
311.23/291.54	        (7) BOUNDS(n^1, INF)
311.23/291.54	    (8) TRS for Loop Detection
311.23/291.54	
311.23/291.54	
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(0)
311.23/291.54	Obligation:
311.23/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.23/291.54	
311.23/291.54	
311.23/291.54	The TRS R consists of the following rules:
311.23/291.54	
311.23/291.54	   div(x, s(y)) -> d(x, s(y), 0)
311.23/291.54	   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
311.23/291.54	   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
311.23/291.54	   cond(false, x, y, z) -> 0
311.23/291.54	   ge(u, 0) -> true
311.23/291.54	   ge(0, s(v)) -> false
311.23/291.54	   ge(s(u), s(v)) -> ge(u, v)
311.23/291.54	   plus(n, 0) -> n
311.23/291.54	   plus(n, s(m)) -> s(plus(n, m))
311.23/291.54	
311.23/291.54	S is empty.
311.23/291.54	Rewrite Strategy: FULL
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
311.23/291.54	Transformed a relative TRS into a decreasing-loop problem.
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(2)
311.23/291.54	Obligation:
311.23/291.54	Analyzing the following TRS for decreasing loops:
311.23/291.54	
311.23/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.23/291.54	
311.23/291.54	
311.23/291.54	The TRS R consists of the following rules:
311.23/291.54	
311.23/291.54	   div(x, s(y)) -> d(x, s(y), 0)
311.23/291.54	   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
311.23/291.54	   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
311.23/291.54	   cond(false, x, y, z) -> 0
311.23/291.54	   ge(u, 0) -> true
311.23/291.54	   ge(0, s(v)) -> false
311.23/291.54	   ge(s(u), s(v)) -> ge(u, v)
311.23/291.54	   plus(n, 0) -> n
311.23/291.54	   plus(n, s(m)) -> s(plus(n, m))
311.23/291.54	
311.23/291.54	S is empty.
311.23/291.54	Rewrite Strategy: FULL
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
311.23/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
311.23/291.54	
311.23/291.54	The rewrite sequence
311.23/291.54	
311.23/291.54	plus(n, s(m)) ->^+ s(plus(n, m))
311.23/291.54	
311.23/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
311.23/291.54	
311.23/291.54	The pumping substitution is [m / s(m)].
311.23/291.54	
311.23/291.54	The result substitution is [ ].
311.23/291.54	
311.23/291.54	
311.23/291.54	
311.23/291.54	
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(4)
311.23/291.54	Complex Obligation (BEST)
311.23/291.54	
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(5)
311.23/291.54	Obligation:
311.23/291.54	Proved the lower bound n^1 for the following obligation:
311.23/291.54	
311.23/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.23/291.54	
311.23/291.54	
311.23/291.54	The TRS R consists of the following rules:
311.23/291.54	
311.23/291.54	   div(x, s(y)) -> d(x, s(y), 0)
311.23/291.54	   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
311.23/291.54	   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
311.23/291.54	   cond(false, x, y, z) -> 0
311.23/291.54	   ge(u, 0) -> true
311.23/291.54	   ge(0, s(v)) -> false
311.23/291.54	   ge(s(u), s(v)) -> ge(u, v)
311.23/291.54	   plus(n, 0) -> n
311.23/291.54	   plus(n, s(m)) -> s(plus(n, m))
311.23/291.54	
311.23/291.54	S is empty.
311.23/291.54	Rewrite Strategy: FULL
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(6) LowerBoundPropagationProof (FINISHED)
311.23/291.54	Propagated lower bound.
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(7)
311.23/291.54	BOUNDS(n^1, INF)
311.23/291.54	
311.23/291.54	----------------------------------------
311.23/291.54	
311.23/291.54	(8)
311.23/291.54	Obligation:
311.23/291.54	Analyzing the following TRS for decreasing loops:
311.23/291.54	
311.23/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.23/291.54	
311.23/291.54	
311.23/291.54	The TRS R consists of the following rules:
311.23/291.54	
311.23/291.54	   div(x, s(y)) -> d(x, s(y), 0)
311.23/291.54	   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
311.23/291.54	   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
311.23/291.54	   cond(false, x, y, z) -> 0
311.23/291.54	   ge(u, 0) -> true
311.23/291.54	   ge(0, s(v)) -> false
311.23/291.54	   ge(s(u), s(v)) -> ge(u, v)
311.23/291.54	   plus(n, 0) -> n
311.23/291.54	   plus(n, s(m)) -> s(plus(n, m))
311.23/291.54	
311.23/291.54	S is empty.
311.23/291.54	Rewrite Strategy: FULL
311.30/291.56	EOF