3.29/1.63	WORST_CASE(NON_POLY, ?)
3.41/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.64	
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.41/1.64	
3.41/1.64	(0) CpxTRS
3.41/1.64	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.41/1.64	(2) TRS for Loop Detection
3.41/1.64	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.41/1.64	(4) BEST
3.41/1.64	    (5) proven lower bound
3.41/1.64	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.41/1.64	        (7) BOUNDS(n^1, INF)
3.41/1.64	    (8) TRS for Loop Detection
3.41/1.64	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.41/1.64	        (10) BOUNDS(INF, INF)
3.41/1.64	
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(0)
3.41/1.64	Obligation:
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   primes -> sieve(from(s(s(0))))
3.41/1.64	   from(X) -> cons(X, from(s(X)))
3.41/1.64	   head(cons(X, Y)) -> X
3.41/1.64	   tail(cons(X, Y)) -> Y
3.41/1.64	   if(true, X, Y) -> X
3.41/1.64	   if(false, X, Y) -> Y
3.41/1.64	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.41/1.64	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.41/1.64	Transformed a relative TRS into a decreasing-loop problem.
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(2)
3.41/1.64	Obligation:
3.41/1.64	Analyzing the following TRS for decreasing loops:
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   primes -> sieve(from(s(s(0))))
3.41/1.64	   from(X) -> cons(X, from(s(X)))
3.41/1.64	   head(cons(X, Y)) -> X
3.41/1.64	   tail(cons(X, Y)) -> Y
3.41/1.64	   if(true, X, Y) -> X
3.41/1.64	   if(false, X, Y) -> Y
3.41/1.64	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.41/1.64	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.41/1.64	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.41/1.64	
3.41/1.64	The rewrite sequence
3.41/1.64	
3.41/1.64	sieve(cons(X, Y)) ->^+ cons(X, filter(X, sieve(Y)))
3.41/1.64	
3.41/1.64	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
3.41/1.64	
3.41/1.64	The pumping substitution is [Y / cons(X, Y)].
3.41/1.64	
3.41/1.64	The result substitution is [ ].
3.41/1.64	
3.41/1.64	
3.41/1.64	
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(4)
3.41/1.64	Complex Obligation (BEST)
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(5)
3.41/1.64	Obligation:
3.41/1.64	Proved the lower bound n^1 for the following obligation:
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   primes -> sieve(from(s(s(0))))
3.41/1.64	   from(X) -> cons(X, from(s(X)))
3.41/1.64	   head(cons(X, Y)) -> X
3.41/1.64	   tail(cons(X, Y)) -> Y
3.41/1.64	   if(true, X, Y) -> X
3.41/1.64	   if(false, X, Y) -> Y
3.41/1.64	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.41/1.64	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(6) LowerBoundPropagationProof (FINISHED)
3.41/1.64	Propagated lower bound.
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(7)
3.41/1.64	BOUNDS(n^1, INF)
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(8)
3.41/1.64	Obligation:
3.41/1.64	Analyzing the following TRS for decreasing loops:
3.41/1.64	
3.41/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.41/1.64	
3.41/1.64	
3.41/1.64	The TRS R consists of the following rules:
3.41/1.64	
3.41/1.64	   primes -> sieve(from(s(s(0))))
3.41/1.64	   from(X) -> cons(X, from(s(X)))
3.41/1.64	   head(cons(X, Y)) -> X
3.41/1.64	   tail(cons(X, Y)) -> Y
3.41/1.64	   if(true, X, Y) -> X
3.41/1.64	   if(false, X, Y) -> Y
3.41/1.64	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.41/1.64	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.41/1.64	
3.41/1.64	S is empty.
3.41/1.64	Rewrite Strategy: INNERMOST
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(9) InfiniteLowerBoundProof (FINISHED)
3.41/1.64	The following loop proves infinite runtime complexity:
3.41/1.64	
3.41/1.64	The rewrite sequence
3.41/1.64	
3.41/1.64	from(X) ->^+ cons(X, from(s(X)))
3.41/1.64	
3.41/1.64	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.41/1.64	
3.41/1.64	The pumping substitution is [ ].
3.41/1.64	
3.41/1.64	The result substitution is [X / s(X)].
3.41/1.64	
3.41/1.64	
3.41/1.64	
3.41/1.64	
3.41/1.64	----------------------------------------
3.41/1.64	
3.41/1.64	(10)
3.41/1.64	BOUNDS(INF, INF)
3.41/1.67	EOF