3.94/1.89	WORST_CASE(NON_POLY, ?)
3.94/1.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.94/1.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.94/1.89	
3.94/1.89	
3.94/1.89	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.94/1.89	
3.94/1.89	(0) CpxTRS
3.94/1.89	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.94/1.89	(2) TRS for Loop Detection
3.94/1.89	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.94/1.89	(4) BEST
3.94/1.89	    (5) proven lower bound
3.94/1.89	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.94/1.89	        (7) BOUNDS(n^1, INF)
3.94/1.89	    (8) TRS for Loop Detection
3.94/1.89	        (9) DecreasingLoopProof [FINISHED, 76 ms]
3.94/1.89	        (10) BOUNDS(EXP, INF)
3.94/1.89	
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(0)
3.94/1.89	Obligation:
3.94/1.89	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.94/1.89	
3.94/1.89	
3.94/1.89	The TRS R consists of the following rules:
3.94/1.89	
3.94/1.89	   a__primes -> a__sieve(a__from(s(s(0))))
3.94/1.89	   a__from(X) -> cons(mark(X), from(s(X)))
3.94/1.89	   a__head(cons(X, Y)) -> mark(X)
3.94/1.89	   a__tail(cons(X, Y)) -> mark(Y)
3.94/1.89	   a__if(true, X, Y) -> mark(X)
3.94/1.89	   a__if(false, X, Y) -> mark(Y)
3.94/1.89	   a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.89	   a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y)))
3.94/1.89	   mark(primes) -> a__primes
3.94/1.89	   mark(sieve(X)) -> a__sieve(mark(X))
3.94/1.89	   mark(from(X)) -> a__from(mark(X))
3.94/1.89	   mark(head(X)) -> a__head(mark(X))
3.94/1.89	   mark(tail(X)) -> a__tail(mark(X))
3.94/1.89	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.94/1.89	   mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2))
3.94/1.89	   mark(s(X)) -> s(mark(X))
3.94/1.89	   mark(0) -> 0
3.94/1.89	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.94/1.89	   mark(true) -> true
3.94/1.89	   mark(false) -> false
3.94/1.89	   mark(divides(X1, X2)) -> divides(mark(X1), mark(X2))
3.94/1.89	   a__primes -> primes
3.94/1.89	   a__sieve(X) -> sieve(X)
3.94/1.89	   a__from(X) -> from(X)
3.94/1.89	   a__head(X) -> head(X)
3.94/1.89	   a__tail(X) -> tail(X)
3.94/1.89	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.94/1.89	   a__filter(X1, X2) -> filter(X1, X2)
3.94/1.89	
3.94/1.89	S is empty.
3.94/1.89	Rewrite Strategy: FULL
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.94/1.89	Transformed a relative TRS into a decreasing-loop problem.
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(2)
3.94/1.89	Obligation:
3.94/1.89	Analyzing the following TRS for decreasing loops:
3.94/1.89	
3.94/1.89	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.94/1.89	
3.94/1.89	
3.94/1.89	The TRS R consists of the following rules:
3.94/1.89	
3.94/1.89	   a__primes -> a__sieve(a__from(s(s(0))))
3.94/1.89	   a__from(X) -> cons(mark(X), from(s(X)))
3.94/1.89	   a__head(cons(X, Y)) -> mark(X)
3.94/1.89	   a__tail(cons(X, Y)) -> mark(Y)
3.94/1.89	   a__if(true, X, Y) -> mark(X)
3.94/1.89	   a__if(false, X, Y) -> mark(Y)
3.94/1.89	   a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.89	   a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y)))
3.94/1.89	   mark(primes) -> a__primes
3.94/1.89	   mark(sieve(X)) -> a__sieve(mark(X))
3.94/1.89	   mark(from(X)) -> a__from(mark(X))
3.94/1.89	   mark(head(X)) -> a__head(mark(X))
3.94/1.89	   mark(tail(X)) -> a__tail(mark(X))
3.94/1.89	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.94/1.89	   mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2))
3.94/1.89	   mark(s(X)) -> s(mark(X))
3.94/1.89	   mark(0) -> 0
3.94/1.89	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.94/1.89	   mark(true) -> true
3.94/1.89	   mark(false) -> false
3.94/1.89	   mark(divides(X1, X2)) -> divides(mark(X1), mark(X2))
3.94/1.89	   a__primes -> primes
3.94/1.89	   a__sieve(X) -> sieve(X)
3.94/1.89	   a__from(X) -> from(X)
3.94/1.89	   a__head(X) -> head(X)
3.94/1.89	   a__tail(X) -> tail(X)
3.94/1.89	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.94/1.89	   a__filter(X1, X2) -> filter(X1, X2)
3.94/1.89	
3.94/1.89	S is empty.
3.94/1.89	Rewrite Strategy: FULL
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.94/1.89	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.94/1.89	
3.94/1.89	The rewrite sequence
3.94/1.89	
3.94/1.89	mark(from(X)) ->^+ a__from(mark(X))
3.94/1.89	
3.94/1.89	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.94/1.89	
3.94/1.89	The pumping substitution is [X / from(X)].
3.94/1.89	
3.94/1.89	The result substitution is [ ].
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(4)
3.94/1.89	Complex Obligation (BEST)
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(5)
3.94/1.89	Obligation:
3.94/1.89	Proved the lower bound n^1 for the following obligation:
3.94/1.89	
3.94/1.89	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.94/1.89	
3.94/1.89	
3.94/1.89	The TRS R consists of the following rules:
3.94/1.89	
3.94/1.89	   a__primes -> a__sieve(a__from(s(s(0))))
3.94/1.89	   a__from(X) -> cons(mark(X), from(s(X)))
3.94/1.89	   a__head(cons(X, Y)) -> mark(X)
3.94/1.89	   a__tail(cons(X, Y)) -> mark(Y)
3.94/1.89	   a__if(true, X, Y) -> mark(X)
3.94/1.89	   a__if(false, X, Y) -> mark(Y)
3.94/1.89	   a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.89	   a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y)))
3.94/1.89	   mark(primes) -> a__primes
3.94/1.89	   mark(sieve(X)) -> a__sieve(mark(X))
3.94/1.89	   mark(from(X)) -> a__from(mark(X))
3.94/1.89	   mark(head(X)) -> a__head(mark(X))
3.94/1.89	   mark(tail(X)) -> a__tail(mark(X))
3.94/1.89	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.94/1.89	   mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2))
3.94/1.89	   mark(s(X)) -> s(mark(X))
3.94/1.89	   mark(0) -> 0
3.94/1.89	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.94/1.89	   mark(true) -> true
3.94/1.89	   mark(false) -> false
3.94/1.89	   mark(divides(X1, X2)) -> divides(mark(X1), mark(X2))
3.94/1.89	   a__primes -> primes
3.94/1.89	   a__sieve(X) -> sieve(X)
3.94/1.89	   a__from(X) -> from(X)
3.94/1.89	   a__head(X) -> head(X)
3.94/1.89	   a__tail(X) -> tail(X)
3.94/1.89	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.94/1.89	   a__filter(X1, X2) -> filter(X1, X2)
3.94/1.89	
3.94/1.89	S is empty.
3.94/1.89	Rewrite Strategy: FULL
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(6) LowerBoundPropagationProof (FINISHED)
3.94/1.89	Propagated lower bound.
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(7)
3.94/1.89	BOUNDS(n^1, INF)
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(8)
3.94/1.89	Obligation:
3.94/1.89	Analyzing the following TRS for decreasing loops:
3.94/1.89	
3.94/1.89	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.94/1.89	
3.94/1.89	
3.94/1.89	The TRS R consists of the following rules:
3.94/1.89	
3.94/1.89	   a__primes -> a__sieve(a__from(s(s(0))))
3.94/1.89	   a__from(X) -> cons(mark(X), from(s(X)))
3.94/1.89	   a__head(cons(X, Y)) -> mark(X)
3.94/1.89	   a__tail(cons(X, Y)) -> mark(Y)
3.94/1.89	   a__if(true, X, Y) -> mark(X)
3.94/1.89	   a__if(false, X, Y) -> mark(Y)
3.94/1.89	   a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.89	   a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y)))
3.94/1.89	   mark(primes) -> a__primes
3.94/1.89	   mark(sieve(X)) -> a__sieve(mark(X))
3.94/1.89	   mark(from(X)) -> a__from(mark(X))
3.94/1.89	   mark(head(X)) -> a__head(mark(X))
3.94/1.89	   mark(tail(X)) -> a__tail(mark(X))
3.94/1.89	   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
3.94/1.89	   mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2))
3.94/1.89	   mark(s(X)) -> s(mark(X))
3.94/1.89	   mark(0) -> 0
3.94/1.89	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.94/1.89	   mark(true) -> true
3.94/1.89	   mark(false) -> false
3.94/1.89	   mark(divides(X1, X2)) -> divides(mark(X1), mark(X2))
3.94/1.89	   a__primes -> primes
3.94/1.89	   a__sieve(X) -> sieve(X)
3.94/1.89	   a__from(X) -> from(X)
3.94/1.89	   a__head(X) -> head(X)
3.94/1.89	   a__tail(X) -> tail(X)
3.94/1.89	   a__if(X1, X2, X3) -> if(X1, X2, X3)
3.94/1.89	   a__filter(X1, X2) -> filter(X1, X2)
3.94/1.89	
3.94/1.89	S is empty.
3.94/1.89	Rewrite Strategy: FULL
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(9) DecreasingLoopProof (FINISHED)
3.94/1.89	The following loop(s) give(s) rise to the lower bound EXP:
3.94/1.89	
3.94/1.89	The rewrite sequence
3.94/1.89	
3.94/1.89	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.94/1.89	
3.94/1.89	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.94/1.89	
3.94/1.89	The pumping substitution is [X / from(X)].
3.94/1.89	
3.94/1.89	The result substitution is [ ].
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	The rewrite sequence
3.94/1.89	
3.94/1.89	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.94/1.89	
3.94/1.89	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.94/1.89	
3.94/1.89	The pumping substitution is [X / from(X)].
3.94/1.89	
3.94/1.89	The result substitution is [ ].
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(10)
3.94/1.89	BOUNDS(EXP, INF)
4.28/3.00	EOF