1143.29/291.60	WORST_CASE(Omega(n^1), ?)
1153.87/294.25	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1153.87/294.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.87/294.25	
1153.87/294.25	(0) CpxTRS
1153.87/294.25	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1153.87/294.25	(2) TRS for Loop Detection
1153.87/294.25	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1153.87/294.25	(4) BEST
1153.87/294.25	    (5) proven lower bound
1153.87/294.25	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1153.87/294.25	        (7) BOUNDS(n^1, INF)
1153.87/294.25	    (8) TRS for Loop Detection
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(0)
1153.87/294.25	Obligation:
1153.87/294.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	The TRS R consists of the following rules:
1153.87/294.25	
1153.87/294.25	   fib(N) -> sel(N, fib1(s(0), s(0)))
1153.87/294.25	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
1153.87/294.25	   add(0, X) -> X
1153.87/294.25	   add(s(X), Y) -> s(add(X, Y))
1153.87/294.25	   sel(0, cons(X, XS)) -> X
1153.87/294.25	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1153.87/294.25	   fib1(X1, X2) -> n__fib1(X1, X2)
1153.87/294.25	   add(X1, X2) -> n__add(X1, X2)
1153.87/294.25	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
1153.87/294.25	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
1153.87/294.25	   activate(X) -> X
1153.87/294.25	
1153.87/294.25	S is empty.
1153.87/294.25	Rewrite Strategy: INNERMOST
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1153.87/294.25	Transformed a relative TRS into a decreasing-loop problem.
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(2)
1153.87/294.25	Obligation:
1153.87/294.25	Analyzing the following TRS for decreasing loops:
1153.87/294.25	
1153.87/294.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	The TRS R consists of the following rules:
1153.87/294.25	
1153.87/294.25	   fib(N) -> sel(N, fib1(s(0), s(0)))
1153.87/294.25	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
1153.87/294.25	   add(0, X) -> X
1153.87/294.25	   add(s(X), Y) -> s(add(X, Y))
1153.87/294.25	   sel(0, cons(X, XS)) -> X
1153.87/294.25	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1153.87/294.25	   fib1(X1, X2) -> n__fib1(X1, X2)
1153.87/294.25	   add(X1, X2) -> n__add(X1, X2)
1153.87/294.25	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
1153.87/294.25	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
1153.87/294.25	   activate(X) -> X
1153.87/294.25	
1153.87/294.25	S is empty.
1153.87/294.25	Rewrite Strategy: INNERMOST
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(3) DecreasingLoopProof (LOWER BOUND(ID))
1153.87/294.25	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1153.87/294.25	
1153.87/294.25	The rewrite sequence
1153.87/294.25	
1153.87/294.25	activate(n__add(X1, X2)) ->^+ add(activate(X1), activate(X2))
1153.87/294.25	
1153.87/294.25	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1153.87/294.25	
1153.87/294.25	The pumping substitution is [X1 / n__add(X1, X2)].
1153.87/294.25	
1153.87/294.25	The result substitution is [ ].
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(4)
1153.87/294.25	Complex Obligation (BEST)
1153.87/294.25	
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(5)
1153.87/294.25	Obligation:
1153.87/294.25	Proved the lower bound n^1 for the following obligation:
1153.87/294.25	
1153.87/294.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	The TRS R consists of the following rules:
1153.87/294.25	
1153.87/294.25	   fib(N) -> sel(N, fib1(s(0), s(0)))
1153.87/294.25	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
1153.87/294.25	   add(0, X) -> X
1153.87/294.25	   add(s(X), Y) -> s(add(X, Y))
1153.87/294.25	   sel(0, cons(X, XS)) -> X
1153.87/294.25	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1153.87/294.25	   fib1(X1, X2) -> n__fib1(X1, X2)
1153.87/294.25	   add(X1, X2) -> n__add(X1, X2)
1153.87/294.25	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
1153.87/294.25	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
1153.87/294.25	   activate(X) -> X
1153.87/294.25	
1153.87/294.25	S is empty.
1153.87/294.25	Rewrite Strategy: INNERMOST
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(6) LowerBoundPropagationProof (FINISHED)
1153.87/294.25	Propagated lower bound.
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(7)
1153.87/294.25	BOUNDS(n^1, INF)
1153.87/294.25	
1153.87/294.25	----------------------------------------
1153.87/294.25	
1153.87/294.25	(8)
1153.87/294.25	Obligation:
1153.87/294.25	Analyzing the following TRS for decreasing loops:
1153.87/294.25	
1153.87/294.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1153.87/294.25	
1153.87/294.25	
1153.87/294.25	The TRS R consists of the following rules:
1153.87/294.25	
1153.87/294.25	   fib(N) -> sel(N, fib1(s(0), s(0)))
1153.87/294.25	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
1153.87/294.25	   add(0, X) -> X
1153.87/294.25	   add(s(X), Y) -> s(add(X, Y))
1153.87/294.25	   sel(0, cons(X, XS)) -> X
1153.87/294.25	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1153.87/294.25	   fib1(X1, X2) -> n__fib1(X1, X2)
1153.87/294.25	   add(X1, X2) -> n__add(X1, X2)
1153.87/294.25	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
1153.87/294.25	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
1153.87/294.25	   activate(X) -> X
1153.87/294.25	
1153.87/294.25	S is empty.
1153.87/294.25	Rewrite Strategy: INNERMOST
1153.87/294.31	EOF