311.13/291.54	WORST_CASE(Omega(n^1), ?)
311.13/291.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
311.13/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.13/291.55	
311.13/291.55	
311.13/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.13/291.55	
311.13/291.55	(0) CpxTRS
311.13/291.55	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
311.13/291.55	(2) TRS for Loop Detection
311.13/291.55	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
311.13/291.55	(4) BEST
311.13/291.55	    (5) proven lower bound
311.13/291.55	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
311.13/291.55	        (7) BOUNDS(n^1, INF)
311.13/291.55	    (8) TRS for Loop Detection
311.13/291.55	
311.13/291.55	
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(0)
311.13/291.55	Obligation:
311.13/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.13/291.55	
311.13/291.55	
311.13/291.55	The TRS R consists of the following rules:
311.13/291.55	
311.13/291.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
311.13/291.55	   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
311.13/291.55	   add(0, X) -> X
311.13/291.55	   add(s(X), Y) -> s(add(X, Y))
311.13/291.55	   sel(0, cons(X, XS)) -> X
311.13/291.55	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
311.13/291.55	   fib1(X1, X2) -> n__fib1(X1, X2)
311.13/291.55	   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
311.13/291.55	   activate(X) -> X
311.13/291.55	
311.13/291.55	S is empty.
311.13/291.55	Rewrite Strategy: FULL
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
311.13/291.55	Transformed a relative TRS into a decreasing-loop problem.
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(2)
311.13/291.55	Obligation:
311.13/291.55	Analyzing the following TRS for decreasing loops:
311.13/291.55	
311.13/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.13/291.55	
311.13/291.55	
311.13/291.55	The TRS R consists of the following rules:
311.13/291.55	
311.13/291.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
311.13/291.55	   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
311.13/291.55	   add(0, X) -> X
311.13/291.55	   add(s(X), Y) -> s(add(X, Y))
311.13/291.55	   sel(0, cons(X, XS)) -> X
311.13/291.55	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
311.13/291.55	   fib1(X1, X2) -> n__fib1(X1, X2)
311.13/291.55	   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
311.13/291.55	   activate(X) -> X
311.13/291.55	
311.13/291.55	S is empty.
311.13/291.55	Rewrite Strategy: FULL
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(3) DecreasingLoopProof (LOWER BOUND(ID))
311.13/291.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
311.13/291.55	
311.13/291.55	The rewrite sequence
311.13/291.55	
311.13/291.55	add(s(X), Y) ->^+ s(add(X, Y))
311.13/291.55	
311.13/291.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
311.13/291.55	
311.13/291.55	The pumping substitution is [X / s(X)].
311.13/291.55	
311.13/291.55	The result substitution is [ ].
311.13/291.55	
311.13/291.55	
311.13/291.55	
311.13/291.55	
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(4)
311.13/291.55	Complex Obligation (BEST)
311.13/291.55	
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(5)
311.13/291.55	Obligation:
311.13/291.55	Proved the lower bound n^1 for the following obligation:
311.13/291.55	
311.13/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.13/291.55	
311.13/291.55	
311.13/291.55	The TRS R consists of the following rules:
311.13/291.55	
311.13/291.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
311.13/291.55	   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
311.13/291.55	   add(0, X) -> X
311.13/291.55	   add(s(X), Y) -> s(add(X, Y))
311.13/291.55	   sel(0, cons(X, XS)) -> X
311.13/291.55	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
311.13/291.55	   fib1(X1, X2) -> n__fib1(X1, X2)
311.13/291.55	   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
311.13/291.55	   activate(X) -> X
311.13/291.55	
311.13/291.55	S is empty.
311.13/291.55	Rewrite Strategy: FULL
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(6) LowerBoundPropagationProof (FINISHED)
311.13/291.55	Propagated lower bound.
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(7)
311.13/291.55	BOUNDS(n^1, INF)
311.13/291.55	
311.13/291.55	----------------------------------------
311.13/291.55	
311.13/291.55	(8)
311.13/291.55	Obligation:
311.13/291.55	Analyzing the following TRS for decreasing loops:
311.13/291.55	
311.13/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.13/291.55	
311.13/291.55	
311.13/291.55	The TRS R consists of the following rules:
311.13/291.55	
311.13/291.55	   fib(N) -> sel(N, fib1(s(0), s(0)))
311.13/291.55	   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
311.13/291.55	   add(0, X) -> X
311.13/291.55	   add(s(X), Y) -> s(add(X, Y))
311.13/291.55	   sel(0, cons(X, XS)) -> X
311.13/291.55	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
311.13/291.55	   fib1(X1, X2) -> n__fib1(X1, X2)
311.13/291.55	   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
311.13/291.55	   activate(X) -> X
311.13/291.55	
311.13/291.55	S is empty.
311.13/291.55	Rewrite Strategy: FULL
311.13/291.58	EOF