3.08/1.60	WORST_CASE(NON_POLY, ?)
3.08/1.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.08/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.08/1.61	
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	(0) CpxTRS
3.08/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.08/1.61	(2) TRS for Loop Detection
3.08/1.61	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.08/1.61	(4) BEST
3.08/1.61	    (5) proven lower bound
3.08/1.61	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.08/1.61	        (7) BOUNDS(n^1, INF)
3.08/1.61	    (8) TRS for Loop Detection
3.08/1.61	        (9) DecreasingLoopProof [FINISHED, 19 ms]
3.08/1.61	        (10) BOUNDS(EXP, INF)
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(0)
3.08/1.61	Obligation:
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
3.08/1.61	   sqr(0) -> 0
3.08/1.61	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
3.08/1.61	   dbl(0) -> 0
3.08/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.08/1.61	   add(0, X) -> X
3.08/1.61	   add(s(X), Y) -> s(add(X, Y))
3.08/1.61	   first(0, X) -> nil
3.08/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.08/1.61	   half(0) -> 0
3.08/1.61	   half(s(0)) -> 0
3.08/1.61	   half(s(s(X))) -> s(half(X))
3.08/1.61	   half(dbl(X)) -> X
3.08/1.61	   terms(X) -> n__terms(X)
3.08/1.61	   s(X) -> n__s(X)
3.08/1.61	   first(X1, X2) -> n__first(X1, X2)
3.08/1.61	   activate(n__terms(X)) -> terms(activate(X))
3.08/1.61	   activate(n__s(X)) -> s(activate(X))
3.08/1.61	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.08/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(2)
3.08/1.61	Obligation:
3.08/1.61	Analyzing the following TRS for decreasing loops:
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
3.08/1.61	   sqr(0) -> 0
3.08/1.61	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
3.08/1.61	   dbl(0) -> 0
3.08/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.08/1.61	   add(0, X) -> X
3.08/1.61	   add(s(X), Y) -> s(add(X, Y))
3.08/1.61	   first(0, X) -> nil
3.08/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.08/1.61	   half(0) -> 0
3.08/1.61	   half(s(0)) -> 0
3.08/1.61	   half(s(s(X))) -> s(half(X))
3.08/1.61	   half(dbl(X)) -> X
3.08/1.61	   terms(X) -> n__terms(X)
3.08/1.61	   s(X) -> n__s(X)
3.08/1.61	   first(X1, X2) -> n__first(X1, X2)
3.08/1.61	   activate(n__terms(X)) -> terms(activate(X))
3.08/1.61	   activate(n__s(X)) -> s(activate(X))
3.08/1.61	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.08/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.08/1.61	
3.08/1.61	The rewrite sequence
3.08/1.61	
3.08/1.61	activate(n__terms(X)) ->^+ terms(activate(X))
3.08/1.61	
3.08/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.08/1.61	
3.08/1.61	The pumping substitution is [X / n__terms(X)].
3.08/1.61	
3.08/1.61	The result substitution is [ ].
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(4)
3.08/1.61	Complex Obligation (BEST)
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(5)
3.08/1.61	Obligation:
3.08/1.61	Proved the lower bound n^1 for the following obligation:
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
3.08/1.61	   sqr(0) -> 0
3.08/1.61	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
3.08/1.61	   dbl(0) -> 0
3.08/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.08/1.61	   add(0, X) -> X
3.08/1.61	   add(s(X), Y) -> s(add(X, Y))
3.08/1.61	   first(0, X) -> nil
3.08/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.08/1.61	   half(0) -> 0
3.08/1.61	   half(s(0)) -> 0
3.08/1.61	   half(s(s(X))) -> s(half(X))
3.08/1.61	   half(dbl(X)) -> X
3.08/1.61	   terms(X) -> n__terms(X)
3.08/1.61	   s(X) -> n__s(X)
3.08/1.61	   first(X1, X2) -> n__first(X1, X2)
3.08/1.61	   activate(n__terms(X)) -> terms(activate(X))
3.08/1.61	   activate(n__s(X)) -> s(activate(X))
3.08/1.61	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(6) LowerBoundPropagationProof (FINISHED)
3.08/1.61	Propagated lower bound.
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(7)
3.08/1.61	BOUNDS(n^1, INF)
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(8)
3.08/1.61	Obligation:
3.08/1.61	Analyzing the following TRS for decreasing loops:
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
3.08/1.61	   sqr(0) -> 0
3.08/1.61	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
3.08/1.61	   dbl(0) -> 0
3.08/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.08/1.61	   add(0, X) -> X
3.08/1.61	   add(s(X), Y) -> s(add(X, Y))
3.08/1.61	   first(0, X) -> nil
3.08/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.08/1.61	   half(0) -> 0
3.08/1.61	   half(s(0)) -> 0
3.08/1.61	   half(s(s(X))) -> s(half(X))
3.08/1.61	   half(dbl(X)) -> X
3.08/1.61	   terms(X) -> n__terms(X)
3.08/1.61	   s(X) -> n__s(X)
3.08/1.61	   first(X1, X2) -> n__first(X1, X2)
3.08/1.61	   activate(n__terms(X)) -> terms(activate(X))
3.08/1.61	   activate(n__s(X)) -> s(activate(X))
3.08/1.61	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(9) DecreasingLoopProof (FINISHED)
3.08/1.61	The following loop(s) give(s) rise to the lower bound EXP:
3.08/1.61	
3.08/1.61	The rewrite sequence
3.08/1.61	
3.08/1.61	activate(n__terms(X)) ->^+ cons(recip(sqr(activate(X))), n__terms(n__s(activate(X))))
3.08/1.61	
3.08/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
3.08/1.61	
3.08/1.61	The pumping substitution is [X / n__terms(X)].
3.08/1.61	
3.08/1.61	The result substitution is [ ].
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	The rewrite sequence
3.08/1.61	
3.08/1.61	activate(n__terms(X)) ->^+ cons(recip(sqr(activate(X))), n__terms(n__s(activate(X))))
3.08/1.61	
3.08/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.08/1.61	
3.08/1.61	The pumping substitution is [X / n__terms(X)].
3.08/1.61	
3.08/1.61	The result substitution is [ ].
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(10)
3.08/1.61	BOUNDS(EXP, INF)
3.30/1.65	EOF