3.29/1.62	WORST_CASE(NON_POLY, ?)
3.29/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.29/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.29/1.63	
3.29/1.63	
3.29/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.29/1.63	
3.29/1.63	(0) CpxTRS
3.29/1.63	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.29/1.63	(2) TRS for Loop Detection
3.29/1.63	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.29/1.63	(4) BEST
3.29/1.63	    (5) proven lower bound
3.29/1.63	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.29/1.63	        (7) BOUNDS(n^1, INF)
3.29/1.63	    (8) TRS for Loop Detection
3.29/1.63	        (9) DecreasingLoopProof [FINISHED, 12 ms]
3.29/1.63	        (10) BOUNDS(EXP, INF)
3.29/1.63	
3.29/1.63	
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(0)
3.29/1.63	Obligation:
3.29/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.29/1.63	
3.29/1.63	
3.29/1.63	The TRS R consists of the following rules:
3.29/1.63	
3.29/1.63	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.29/1.63	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
3.29/1.63	   add(0, X) -> X
3.29/1.63	   add(s(X), Y) -> s(add(X, Y))
3.29/1.63	   sel(0, cons(X, XS)) -> X
3.29/1.63	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.29/1.63	   fib1(X1, X2) -> n__fib1(X1, X2)
3.29/1.63	   add(X1, X2) -> n__add(X1, X2)
3.29/1.63	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
3.29/1.63	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.29/1.63	   activate(X) -> X
3.29/1.63	
3.29/1.63	S is empty.
3.29/1.63	Rewrite Strategy: FULL
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.29/1.63	Transformed a relative TRS into a decreasing-loop problem.
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(2)
3.29/1.63	Obligation:
3.29/1.63	Analyzing the following TRS for decreasing loops:
3.29/1.63	
3.29/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.29/1.63	
3.29/1.63	
3.29/1.63	The TRS R consists of the following rules:
3.29/1.63	
3.29/1.63	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.29/1.63	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
3.29/1.63	   add(0, X) -> X
3.29/1.63	   add(s(X), Y) -> s(add(X, Y))
3.29/1.63	   sel(0, cons(X, XS)) -> X
3.29/1.63	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.29/1.63	   fib1(X1, X2) -> n__fib1(X1, X2)
3.29/1.63	   add(X1, X2) -> n__add(X1, X2)
3.29/1.63	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
3.29/1.63	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.29/1.63	   activate(X) -> X
3.29/1.63	
3.29/1.63	S is empty.
3.29/1.63	Rewrite Strategy: FULL
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.29/1.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.29/1.63	
3.29/1.63	The rewrite sequence
3.29/1.63	
3.29/1.63	activate(n__add(X1, X2)) ->^+ add(activate(X1), activate(X2))
3.29/1.63	
3.29/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.29/1.63	
3.29/1.63	The pumping substitution is [X1 / n__add(X1, X2)].
3.29/1.63	
3.29/1.63	The result substitution is [ ].
3.29/1.63	
3.29/1.63	
3.29/1.63	
3.29/1.63	
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(4)
3.29/1.63	Complex Obligation (BEST)
3.29/1.63	
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(5)
3.29/1.63	Obligation:
3.29/1.63	Proved the lower bound n^1 for the following obligation:
3.29/1.63	
3.29/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.29/1.63	
3.29/1.63	
3.29/1.63	The TRS R consists of the following rules:
3.29/1.63	
3.29/1.63	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.29/1.63	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
3.29/1.63	   add(0, X) -> X
3.29/1.63	   add(s(X), Y) -> s(add(X, Y))
3.29/1.63	   sel(0, cons(X, XS)) -> X
3.29/1.63	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.29/1.63	   fib1(X1, X2) -> n__fib1(X1, X2)
3.29/1.63	   add(X1, X2) -> n__add(X1, X2)
3.29/1.63	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
3.29/1.63	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.29/1.63	   activate(X) -> X
3.29/1.63	
3.29/1.63	S is empty.
3.29/1.63	Rewrite Strategy: FULL
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(6) LowerBoundPropagationProof (FINISHED)
3.29/1.63	Propagated lower bound.
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(7)
3.29/1.63	BOUNDS(n^1, INF)
3.29/1.63	
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(8)
3.29/1.63	Obligation:
3.29/1.63	Analyzing the following TRS for decreasing loops:
3.29/1.63	
3.29/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.29/1.63	
3.29/1.63	
3.29/1.63	The TRS R consists of the following rules:
3.29/1.63	
3.29/1.63	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.29/1.63	   fib1(X, Y) -> cons(X, n__fib1(Y, n__add(X, Y)))
3.29/1.63	   add(0, X) -> X
3.29/1.63	   add(s(X), Y) -> s(add(X, Y))
3.29/1.63	   sel(0, cons(X, XS)) -> X
3.29/1.63	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.29/1.63	   fib1(X1, X2) -> n__fib1(X1, X2)
3.29/1.63	   add(X1, X2) -> n__add(X1, X2)
3.29/1.63	   activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
3.29/1.63	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.29/1.63	   activate(X) -> X
3.29/1.63	
3.29/1.63	S is empty.
3.29/1.63	Rewrite Strategy: FULL
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(9) DecreasingLoopProof (FINISHED)
3.29/1.63	The following loop(s) give(s) rise to the lower bound EXP:
3.29/1.63	
3.29/1.63	The rewrite sequence
3.29/1.63	
3.29/1.63	activate(n__fib1(X1, X2)) ->^+ cons(activate(X1), n__fib1(activate(X2), n__add(activate(X1), activate(X2))))
3.29/1.63	
3.29/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.29/1.63	
3.29/1.63	The pumping substitution is [X1 / n__fib1(X1, X2)].
3.29/1.63	
3.29/1.63	The result substitution is [ ].
3.29/1.63	
3.29/1.63	
3.29/1.63	
3.29/1.63	The rewrite sequence
3.29/1.63	
3.29/1.63	activate(n__fib1(X1, X2)) ->^+ cons(activate(X1), n__fib1(activate(X2), n__add(activate(X1), activate(X2))))
3.29/1.63	
3.29/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,0].
3.29/1.63	
3.29/1.63	The pumping substitution is [X1 / n__fib1(X1, X2)].
3.29/1.63	
3.29/1.63	The result substitution is [ ].
3.29/1.63	
3.29/1.63	
3.29/1.63	
3.29/1.63	
3.29/1.63	----------------------------------------
3.29/1.63	
3.29/1.63	(10)
3.29/1.63	BOUNDS(EXP, INF)
3.29/1.65	EOF