904.81/291.53	WORST_CASE(Omega(n^1), ?)
904.81/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
904.81/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
904.81/291.54	
904.81/291.54	
904.81/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
904.81/291.54	
904.81/291.54	(0) CpxTRS
904.81/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
904.81/291.54	(2) TRS for Loop Detection
904.81/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
904.81/291.54	(4) BEST
904.81/291.54	    (5) proven lower bound
904.81/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
904.81/291.54	        (7) BOUNDS(n^1, INF)
904.81/291.54	    (8) TRS for Loop Detection
904.81/291.54	
904.81/291.54	
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(0)
904.81/291.54	Obligation:
904.81/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
904.81/291.54	
904.81/291.54	
904.81/291.54	The TRS R consists of the following rules:
904.81/291.54	
904.81/291.54	   fib(0) -> 0
904.81/291.54	   fib(s(0)) -> s(0)
904.81/291.54	   fib(s(s(0))) -> s(0)
904.81/291.54	   fib(s(s(x))) -> sp(g(x))
904.81/291.54	   g(0) -> pair(s(0), 0)
904.81/291.54	   g(s(0)) -> pair(s(0), s(0))
904.81/291.54	   g(s(x)) -> np(g(x))
904.81/291.54	   sp(pair(x, y)) -> +(x, y)
904.81/291.54	   np(pair(x, y)) -> pair(+(x, y), x)
904.81/291.54	   +(x, 0) -> x
904.81/291.54	   +(x, s(y)) -> s(+(x, y))
904.81/291.54	
904.81/291.54	S is empty.
904.81/291.54	Rewrite Strategy: INNERMOST
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
904.81/291.54	Transformed a relative TRS into a decreasing-loop problem.
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(2)
904.81/291.54	Obligation:
904.81/291.54	Analyzing the following TRS for decreasing loops:
904.81/291.54	
904.81/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
904.81/291.54	
904.81/291.54	
904.81/291.54	The TRS R consists of the following rules:
904.81/291.54	
904.81/291.54	   fib(0) -> 0
904.81/291.54	   fib(s(0)) -> s(0)
904.81/291.54	   fib(s(s(0))) -> s(0)
904.81/291.54	   fib(s(s(x))) -> sp(g(x))
904.81/291.54	   g(0) -> pair(s(0), 0)
904.81/291.54	   g(s(0)) -> pair(s(0), s(0))
904.81/291.54	   g(s(x)) -> np(g(x))
904.81/291.54	   sp(pair(x, y)) -> +(x, y)
904.81/291.54	   np(pair(x, y)) -> pair(+(x, y), x)
904.81/291.54	   +(x, 0) -> x
904.81/291.54	   +(x, s(y)) -> s(+(x, y))
904.81/291.54	
904.81/291.54	S is empty.
904.81/291.54	Rewrite Strategy: INNERMOST
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
904.81/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
904.81/291.54	
904.81/291.54	The rewrite sequence
904.81/291.54	
904.81/291.54	+(x, s(y)) ->^+ s(+(x, y))
904.81/291.54	
904.81/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
904.81/291.54	
904.81/291.54	The pumping substitution is [y / s(y)].
904.81/291.54	
904.81/291.54	The result substitution is [ ].
904.81/291.54	
904.81/291.54	
904.81/291.54	
904.81/291.54	
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(4)
904.81/291.54	Complex Obligation (BEST)
904.81/291.54	
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(5)
904.81/291.54	Obligation:
904.81/291.54	Proved the lower bound n^1 for the following obligation:
904.81/291.54	
904.81/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
904.81/291.54	
904.81/291.54	
904.81/291.54	The TRS R consists of the following rules:
904.81/291.54	
904.81/291.54	   fib(0) -> 0
904.81/291.54	   fib(s(0)) -> s(0)
904.81/291.54	   fib(s(s(0))) -> s(0)
904.81/291.54	   fib(s(s(x))) -> sp(g(x))
904.81/291.54	   g(0) -> pair(s(0), 0)
904.81/291.54	   g(s(0)) -> pair(s(0), s(0))
904.81/291.54	   g(s(x)) -> np(g(x))
904.81/291.54	   sp(pair(x, y)) -> +(x, y)
904.81/291.54	   np(pair(x, y)) -> pair(+(x, y), x)
904.81/291.54	   +(x, 0) -> x
904.81/291.54	   +(x, s(y)) -> s(+(x, y))
904.81/291.54	
904.81/291.54	S is empty.
904.81/291.54	Rewrite Strategy: INNERMOST
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(6) LowerBoundPropagationProof (FINISHED)
904.81/291.54	Propagated lower bound.
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(7)
904.81/291.54	BOUNDS(n^1, INF)
904.81/291.54	
904.81/291.54	----------------------------------------
904.81/291.54	
904.81/291.54	(8)
904.81/291.54	Obligation:
904.81/291.54	Analyzing the following TRS for decreasing loops:
904.81/291.54	
904.81/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
904.81/291.54	
904.81/291.54	
904.81/291.54	The TRS R consists of the following rules:
904.81/291.54	
904.81/291.54	   fib(0) -> 0
904.81/291.54	   fib(s(0)) -> s(0)
904.81/291.54	   fib(s(s(0))) -> s(0)
904.81/291.54	   fib(s(s(x))) -> sp(g(x))
904.81/291.54	   g(0) -> pair(s(0), 0)
904.81/291.54	   g(s(0)) -> pair(s(0), s(0))
904.81/291.54	   g(s(x)) -> np(g(x))
904.81/291.54	   sp(pair(x, y)) -> +(x, y)
904.81/291.54	   np(pair(x, y)) -> pair(+(x, y), x)
904.81/291.54	   +(x, 0) -> x
904.81/291.54	   +(x, s(y)) -> s(+(x, y))
904.81/291.54	
904.81/291.54	S is empty.
904.81/291.54	Rewrite Strategy: INNERMOST
904.98/291.58	EOF