3.38/1.67	WORST_CASE(NON_POLY, ?)
3.38/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.38/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.38/1.68	
3.38/1.68	
3.38/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.68	
3.38/1.68	(0) CpxTRS
3.38/1.68	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.38/1.68	(2) TRS for Loop Detection
3.38/1.68	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.38/1.68	(4) BEST
3.38/1.68	    (5) proven lower bound
3.38/1.68	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.38/1.68	        (7) BOUNDS(n^1, INF)
3.38/1.68	    (8) TRS for Loop Detection
3.38/1.68	        (9) DecreasingLoopProof [FINISHED, 22 ms]
3.38/1.68	        (10) BOUNDS(EXP, INF)
3.38/1.68	
3.38/1.68	
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(0)
3.38/1.68	Obligation:
3.38/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.68	
3.38/1.68	
3.38/1.68	The TRS R consists of the following rules:
3.38/1.68	
3.38/1.68	   primes -> sieve(from(s(s(0))))
3.38/1.68	   from(X) -> cons(X, n__from(n__s(X)))
3.38/1.68	   head(cons(X, Y)) -> X
3.38/1.68	   tail(cons(X, Y)) -> activate(Y)
3.38/1.68	   if(true, X, Y) -> activate(X)
3.38/1.68	   if(false, X, Y) -> activate(Y)
3.38/1.68	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
3.38/1.68	   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
3.38/1.68	   from(X) -> n__from(X)
3.38/1.68	   s(X) -> n__s(X)
3.38/1.68	   filter(X1, X2) -> n__filter(X1, X2)
3.38/1.68	   cons(X1, X2) -> n__cons(X1, X2)
3.38/1.68	   sieve(X) -> n__sieve(X)
3.38/1.68	   activate(n__from(X)) -> from(activate(X))
3.38/1.68	   activate(n__s(X)) -> s(activate(X))
3.38/1.68	   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
3.38/1.68	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.38/1.68	   activate(n__sieve(X)) -> sieve(activate(X))
3.38/1.68	   activate(X) -> X
3.38/1.68	
3.38/1.68	S is empty.
3.38/1.68	Rewrite Strategy: FULL
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.38/1.68	Transformed a relative TRS into a decreasing-loop problem.
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(2)
3.38/1.68	Obligation:
3.38/1.68	Analyzing the following TRS for decreasing loops:
3.38/1.68	
3.38/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.68	
3.38/1.68	
3.38/1.68	The TRS R consists of the following rules:
3.38/1.68	
3.38/1.68	   primes -> sieve(from(s(s(0))))
3.38/1.68	   from(X) -> cons(X, n__from(n__s(X)))
3.38/1.68	   head(cons(X, Y)) -> X
3.38/1.68	   tail(cons(X, Y)) -> activate(Y)
3.38/1.68	   if(true, X, Y) -> activate(X)
3.38/1.68	   if(false, X, Y) -> activate(Y)
3.38/1.68	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
3.38/1.68	   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
3.38/1.68	   from(X) -> n__from(X)
3.38/1.68	   s(X) -> n__s(X)
3.38/1.68	   filter(X1, X2) -> n__filter(X1, X2)
3.38/1.68	   cons(X1, X2) -> n__cons(X1, X2)
3.38/1.68	   sieve(X) -> n__sieve(X)
3.38/1.68	   activate(n__from(X)) -> from(activate(X))
3.38/1.68	   activate(n__s(X)) -> s(activate(X))
3.38/1.68	   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
3.38/1.68	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.38/1.68	   activate(n__sieve(X)) -> sieve(activate(X))
3.38/1.68	   activate(X) -> X
3.38/1.68	
3.38/1.68	S is empty.
3.38/1.68	Rewrite Strategy: FULL
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.38/1.68	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.38/1.68	
3.38/1.68	The rewrite sequence
3.38/1.68	
3.38/1.68	activate(n__s(X)) ->^+ s(activate(X))
3.38/1.68	
3.38/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.38/1.68	
3.38/1.68	The pumping substitution is [X / n__s(X)].
3.38/1.68	
3.38/1.68	The result substitution is [ ].
3.38/1.68	
3.38/1.68	
3.38/1.68	
3.38/1.68	
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(4)
3.38/1.68	Complex Obligation (BEST)
3.38/1.68	
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(5)
3.38/1.68	Obligation:
3.38/1.68	Proved the lower bound n^1 for the following obligation:
3.38/1.68	
3.38/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.68	
3.38/1.68	
3.38/1.68	The TRS R consists of the following rules:
3.38/1.68	
3.38/1.68	   primes -> sieve(from(s(s(0))))
3.38/1.68	   from(X) -> cons(X, n__from(n__s(X)))
3.38/1.68	   head(cons(X, Y)) -> X
3.38/1.68	   tail(cons(X, Y)) -> activate(Y)
3.38/1.68	   if(true, X, Y) -> activate(X)
3.38/1.68	   if(false, X, Y) -> activate(Y)
3.38/1.68	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
3.38/1.68	   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
3.38/1.68	   from(X) -> n__from(X)
3.38/1.68	   s(X) -> n__s(X)
3.38/1.68	   filter(X1, X2) -> n__filter(X1, X2)
3.38/1.68	   cons(X1, X2) -> n__cons(X1, X2)
3.38/1.68	   sieve(X) -> n__sieve(X)
3.38/1.68	   activate(n__from(X)) -> from(activate(X))
3.38/1.68	   activate(n__s(X)) -> s(activate(X))
3.38/1.68	   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
3.38/1.68	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.38/1.68	   activate(n__sieve(X)) -> sieve(activate(X))
3.38/1.68	   activate(X) -> X
3.38/1.68	
3.38/1.68	S is empty.
3.38/1.68	Rewrite Strategy: FULL
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(6) LowerBoundPropagationProof (FINISHED)
3.38/1.68	Propagated lower bound.
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(7)
3.38/1.68	BOUNDS(n^1, INF)
3.38/1.68	
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(8)
3.38/1.68	Obligation:
3.38/1.68	Analyzing the following TRS for decreasing loops:
3.38/1.68	
3.38/1.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.68	
3.38/1.68	
3.38/1.68	The TRS R consists of the following rules:
3.38/1.68	
3.38/1.68	   primes -> sieve(from(s(s(0))))
3.38/1.68	   from(X) -> cons(X, n__from(n__s(X)))
3.38/1.68	   head(cons(X, Y)) -> X
3.38/1.68	   tail(cons(X, Y)) -> activate(Y)
3.38/1.68	   if(true, X, Y) -> activate(X)
3.38/1.68	   if(false, X, Y) -> activate(Y)
3.38/1.68	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
3.38/1.68	   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
3.38/1.68	   from(X) -> n__from(X)
3.38/1.68	   s(X) -> n__s(X)
3.38/1.68	   filter(X1, X2) -> n__filter(X1, X2)
3.38/1.68	   cons(X1, X2) -> n__cons(X1, X2)
3.38/1.68	   sieve(X) -> n__sieve(X)
3.38/1.68	   activate(n__from(X)) -> from(activate(X))
3.38/1.68	   activate(n__s(X)) -> s(activate(X))
3.38/1.68	   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
3.38/1.68	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.38/1.68	   activate(n__sieve(X)) -> sieve(activate(X))
3.38/1.68	   activate(X) -> X
3.38/1.68	
3.38/1.68	S is empty.
3.38/1.68	Rewrite Strategy: FULL
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(9) DecreasingLoopProof (FINISHED)
3.38/1.68	The following loop(s) give(s) rise to the lower bound EXP:
3.38/1.68	
3.38/1.68	The rewrite sequence
3.38/1.68	
3.38/1.68	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.38/1.68	
3.38/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.38/1.68	
3.38/1.68	The pumping substitution is [X / n__from(X)].
3.38/1.68	
3.38/1.68	The result substitution is [ ].
3.38/1.68	
3.38/1.68	
3.38/1.68	
3.38/1.68	The rewrite sequence
3.38/1.68	
3.38/1.68	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.38/1.68	
3.38/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.38/1.68	
3.38/1.68	The pumping substitution is [X / n__from(X)].
3.38/1.68	
3.38/1.68	The result substitution is [ ].
3.38/1.68	
3.38/1.68	
3.38/1.68	
3.38/1.68	
3.38/1.68	----------------------------------------
3.38/1.68	
3.38/1.68	(10)
3.38/1.68	BOUNDS(EXP, INF)
3.66/1.72	EOF