3.52/1.69	WORST_CASE(NON_POLY, ?)
3.68/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.68/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.68/1.71	
3.68/1.71	
3.68/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.68/1.71	
3.68/1.71	(0) CpxTRS
3.68/1.71	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.68/1.71	(2) TRS for Loop Detection
3.68/1.71	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.68/1.71	(4) BEST
3.68/1.71	    (5) proven lower bound
3.68/1.71	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.68/1.71	        (7) BOUNDS(n^1, INF)
3.68/1.71	    (8) TRS for Loop Detection
3.68/1.71	        (9) DecreasingLoopProof [FINISHED, 13 ms]
3.68/1.71	        (10) BOUNDS(EXP, INF)
3.68/1.71	
3.68/1.71	
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(0)
3.68/1.71	Obligation:
3.68/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.68/1.71	
3.68/1.71	
3.68/1.71	The TRS R consists of the following rules:
3.68/1.71	
3.68/1.71	   a__fib(N) -> a__sel(mark(N), a__fib1(s(0), s(0)))
3.68/1.71	   a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
3.68/1.71	   a__add(0, X) -> mark(X)
3.68/1.71	   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
3.68/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.68/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.68/1.71	   mark(fib(X)) -> a__fib(mark(X))
3.68/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.68/1.71	   mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
3.68/1.71	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.68/1.71	   mark(s(X)) -> s(mark(X))
3.68/1.71	   mark(0) -> 0
3.68/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.68/1.71	   a__fib(X) -> fib(X)
3.68/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.68/1.71	   a__fib1(X1, X2) -> fib1(X1, X2)
3.68/1.71	   a__add(X1, X2) -> add(X1, X2)
3.68/1.71	
3.68/1.71	S is empty.
3.68/1.71	Rewrite Strategy: FULL
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.68/1.71	Transformed a relative TRS into a decreasing-loop problem.
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(2)
3.68/1.71	Obligation:
3.68/1.71	Analyzing the following TRS for decreasing loops:
3.68/1.71	
3.68/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.68/1.71	
3.68/1.71	
3.68/1.71	The TRS R consists of the following rules:
3.68/1.71	
3.68/1.71	   a__fib(N) -> a__sel(mark(N), a__fib1(s(0), s(0)))
3.68/1.71	   a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
3.68/1.71	   a__add(0, X) -> mark(X)
3.68/1.71	   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
3.68/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.68/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.68/1.71	   mark(fib(X)) -> a__fib(mark(X))
3.68/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.68/1.71	   mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
3.68/1.71	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.68/1.71	   mark(s(X)) -> s(mark(X))
3.68/1.71	   mark(0) -> 0
3.68/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.68/1.71	   a__fib(X) -> fib(X)
3.68/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.68/1.71	   a__fib1(X1, X2) -> fib1(X1, X2)
3.68/1.71	   a__add(X1, X2) -> add(X1, X2)
3.68/1.71	
3.68/1.71	S is empty.
3.68/1.71	Rewrite Strategy: FULL
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.68/1.71	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.68/1.71	
3.68/1.71	The rewrite sequence
3.68/1.71	
3.68/1.71	mark(fib1(X1, X2)) ->^+ a__fib1(mark(X1), mark(X2))
3.68/1.71	
3.68/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.68/1.71	
3.68/1.71	The pumping substitution is [X1 / fib1(X1, X2)].
3.68/1.71	
3.68/1.71	The result substitution is [ ].
3.68/1.71	
3.68/1.71	
3.68/1.71	
3.68/1.71	
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(4)
3.68/1.71	Complex Obligation (BEST)
3.68/1.71	
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(5)
3.68/1.71	Obligation:
3.68/1.71	Proved the lower bound n^1 for the following obligation:
3.68/1.71	
3.68/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.68/1.71	
3.68/1.71	
3.68/1.71	The TRS R consists of the following rules:
3.68/1.71	
3.68/1.71	   a__fib(N) -> a__sel(mark(N), a__fib1(s(0), s(0)))
3.68/1.71	   a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
3.68/1.71	   a__add(0, X) -> mark(X)
3.68/1.71	   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
3.68/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.68/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.68/1.71	   mark(fib(X)) -> a__fib(mark(X))
3.68/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.68/1.71	   mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
3.68/1.71	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.68/1.71	   mark(s(X)) -> s(mark(X))
3.68/1.71	   mark(0) -> 0
3.68/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.68/1.71	   a__fib(X) -> fib(X)
3.68/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.68/1.71	   a__fib1(X1, X2) -> fib1(X1, X2)
3.68/1.71	   a__add(X1, X2) -> add(X1, X2)
3.68/1.71	
3.68/1.71	S is empty.
3.68/1.71	Rewrite Strategy: FULL
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(6) LowerBoundPropagationProof (FINISHED)
3.68/1.71	Propagated lower bound.
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(7)
3.68/1.71	BOUNDS(n^1, INF)
3.68/1.71	
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(8)
3.68/1.71	Obligation:
3.68/1.71	Analyzing the following TRS for decreasing loops:
3.68/1.71	
3.68/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.68/1.71	
3.68/1.71	
3.68/1.71	The TRS R consists of the following rules:
3.68/1.71	
3.68/1.71	   a__fib(N) -> a__sel(mark(N), a__fib1(s(0), s(0)))
3.68/1.71	   a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
3.68/1.71	   a__add(0, X) -> mark(X)
3.68/1.71	   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
3.68/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.68/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.68/1.71	   mark(fib(X)) -> a__fib(mark(X))
3.68/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.68/1.71	   mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
3.68/1.71	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.68/1.71	   mark(s(X)) -> s(mark(X))
3.68/1.71	   mark(0) -> 0
3.68/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.68/1.71	   a__fib(X) -> fib(X)
3.68/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.68/1.71	   a__fib1(X1, X2) -> fib1(X1, X2)
3.68/1.71	   a__add(X1, X2) -> add(X1, X2)
3.68/1.71	
3.68/1.71	S is empty.
3.68/1.71	Rewrite Strategy: FULL
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(9) DecreasingLoopProof (FINISHED)
3.68/1.71	The following loop(s) give(s) rise to the lower bound EXP:
3.68/1.71	
3.68/1.71	The rewrite sequence
3.68/1.71	
3.68/1.71	mark(fib1(X1, X2)) ->^+ cons(mark(mark(X1)), fib1(mark(X2), add(mark(X1), mark(X2))))
3.68/1.71	
3.68/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.68/1.71	
3.68/1.71	The pumping substitution is [X1 / fib1(X1, X2)].
3.68/1.71	
3.68/1.71	The result substitution is [ ].
3.68/1.71	
3.68/1.71	
3.68/1.71	
3.68/1.71	The rewrite sequence
3.68/1.71	
3.68/1.71	mark(fib1(X1, X2)) ->^+ cons(mark(mark(X1)), fib1(mark(X2), add(mark(X1), mark(X2))))
3.68/1.71	
3.68/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,0].
3.68/1.71	
3.68/1.71	The pumping substitution is [X1 / fib1(X1, X2)].
3.68/1.71	
3.68/1.71	The result substitution is [ ].
3.68/1.71	
3.68/1.71	
3.68/1.71	
3.68/1.71	
3.68/1.71	----------------------------------------
3.68/1.71	
3.68/1.71	(10)
3.68/1.71	BOUNDS(EXP, INF)
3.71/2.69	EOF