3.17/2.03	WORST_CASE(NON_POLY, ?)
3.17/2.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.17/2.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.17/2.04	
3.17/2.04	
3.17/2.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.17/2.04	
3.17/2.04	(0) CpxTRS
3.17/2.04	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.17/2.04	(2) TRS for Loop Detection
3.17/2.04	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.17/2.04	(4) BEST
3.17/2.04	    (5) proven lower bound
3.17/2.04	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.17/2.04	        (7) BOUNDS(n^1, INF)
3.17/2.04	    (8) TRS for Loop Detection
3.17/2.04	        (9) DecreasingLoopProof [FINISHED, 25 ms]
3.17/2.04	        (10) BOUNDS(EXP, INF)
3.17/2.04	
3.17/2.04	
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(0)
3.17/2.04	Obligation:
3.17/2.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.17/2.04	
3.17/2.04	
3.17/2.04	The TRS R consists of the following rules:
3.17/2.04	
3.17/2.04	   from(X) -> cons(X, n__from(n__s(X)))
3.17/2.04	   2ndspos(0, Z) -> rnil
3.17/2.04	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
3.17/2.04	   2ndsneg(0, Z) -> rnil
3.17/2.04	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
3.17/2.04	   pi(X) -> 2ndspos(X, from(0))
3.17/2.04	   plus(0, Y) -> Y
3.17/2.04	   plus(s(X), Y) -> s(plus(X, Y))
3.17/2.04	   times(0, Y) -> 0
3.17/2.04	   times(s(X), Y) -> plus(Y, times(X, Y))
3.17/2.04	   square(X) -> times(X, X)
3.17/2.04	   from(X) -> n__from(X)
3.17/2.04	   s(X) -> n__s(X)
3.17/2.04	   activate(n__from(X)) -> from(activate(X))
3.17/2.04	   activate(n__s(X)) -> s(activate(X))
3.17/2.04	   activate(X) -> X
3.17/2.04	
3.17/2.04	S is empty.
3.17/2.04	Rewrite Strategy: FULL
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.17/2.04	Transformed a relative TRS into a decreasing-loop problem.
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(2)
3.17/2.04	Obligation:
3.17/2.04	Analyzing the following TRS for decreasing loops:
3.17/2.04	
3.17/2.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.17/2.04	
3.17/2.04	
3.17/2.04	The TRS R consists of the following rules:
3.17/2.04	
3.17/2.04	   from(X) -> cons(X, n__from(n__s(X)))
3.17/2.04	   2ndspos(0, Z) -> rnil
3.17/2.04	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
3.17/2.04	   2ndsneg(0, Z) -> rnil
3.17/2.04	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
3.17/2.04	   pi(X) -> 2ndspos(X, from(0))
3.17/2.04	   plus(0, Y) -> Y
3.17/2.04	   plus(s(X), Y) -> s(plus(X, Y))
3.17/2.04	   times(0, Y) -> 0
3.17/2.04	   times(s(X), Y) -> plus(Y, times(X, Y))
3.17/2.04	   square(X) -> times(X, X)
3.17/2.04	   from(X) -> n__from(X)
3.17/2.04	   s(X) -> n__s(X)
3.17/2.04	   activate(n__from(X)) -> from(activate(X))
3.17/2.04	   activate(n__s(X)) -> s(activate(X))
3.17/2.04	   activate(X) -> X
3.17/2.04	
3.17/2.04	S is empty.
3.17/2.04	Rewrite Strategy: FULL
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.17/2.04	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.17/2.04	
3.17/2.04	The rewrite sequence
3.17/2.04	
3.17/2.04	activate(n__s(X)) ->^+ s(activate(X))
3.17/2.04	
3.17/2.04	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.17/2.04	
3.17/2.04	The pumping substitution is [X / n__s(X)].
3.17/2.04	
3.17/2.04	The result substitution is [ ].
3.17/2.04	
3.17/2.04	
3.17/2.04	
3.17/2.04	
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(4)
3.17/2.04	Complex Obligation (BEST)
3.17/2.04	
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(5)
3.17/2.04	Obligation:
3.17/2.04	Proved the lower bound n^1 for the following obligation:
3.17/2.04	
3.17/2.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.17/2.04	
3.17/2.04	
3.17/2.04	The TRS R consists of the following rules:
3.17/2.04	
3.17/2.04	   from(X) -> cons(X, n__from(n__s(X)))
3.17/2.04	   2ndspos(0, Z) -> rnil
3.17/2.04	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
3.17/2.04	   2ndsneg(0, Z) -> rnil
3.17/2.04	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
3.17/2.04	   pi(X) -> 2ndspos(X, from(0))
3.17/2.04	   plus(0, Y) -> Y
3.17/2.04	   plus(s(X), Y) -> s(plus(X, Y))
3.17/2.04	   times(0, Y) -> 0
3.17/2.04	   times(s(X), Y) -> plus(Y, times(X, Y))
3.17/2.04	   square(X) -> times(X, X)
3.17/2.04	   from(X) -> n__from(X)
3.17/2.04	   s(X) -> n__s(X)
3.17/2.04	   activate(n__from(X)) -> from(activate(X))
3.17/2.04	   activate(n__s(X)) -> s(activate(X))
3.17/2.04	   activate(X) -> X
3.17/2.04	
3.17/2.04	S is empty.
3.17/2.04	Rewrite Strategy: FULL
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(6) LowerBoundPropagationProof (FINISHED)
3.17/2.04	Propagated lower bound.
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(7)
3.17/2.04	BOUNDS(n^1, INF)
3.17/2.04	
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(8)
3.17/2.04	Obligation:
3.17/2.04	Analyzing the following TRS for decreasing loops:
3.17/2.04	
3.17/2.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.17/2.04	
3.17/2.04	
3.17/2.04	The TRS R consists of the following rules:
3.17/2.04	
3.17/2.04	   from(X) -> cons(X, n__from(n__s(X)))
3.17/2.04	   2ndspos(0, Z) -> rnil
3.17/2.04	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
3.17/2.04	   2ndsneg(0, Z) -> rnil
3.17/2.04	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
3.17/2.04	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
3.17/2.04	   pi(X) -> 2ndspos(X, from(0))
3.17/2.04	   plus(0, Y) -> Y
3.17/2.04	   plus(s(X), Y) -> s(plus(X, Y))
3.17/2.04	   times(0, Y) -> 0
3.17/2.04	   times(s(X), Y) -> plus(Y, times(X, Y))
3.17/2.04	   square(X) -> times(X, X)
3.17/2.04	   from(X) -> n__from(X)
3.17/2.04	   s(X) -> n__s(X)
3.17/2.04	   activate(n__from(X)) -> from(activate(X))
3.17/2.04	   activate(n__s(X)) -> s(activate(X))
3.17/2.04	   activate(X) -> X
3.17/2.04	
3.17/2.04	S is empty.
3.17/2.04	Rewrite Strategy: FULL
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(9) DecreasingLoopProof (FINISHED)
3.17/2.04	The following loop(s) give(s) rise to the lower bound EXP:
3.17/2.04	
3.17/2.04	The rewrite sequence
3.17/2.04	
3.17/2.04	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.17/2.04	
3.17/2.04	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.17/2.04	
3.17/2.04	The pumping substitution is [X / n__from(X)].
3.17/2.04	
3.17/2.04	The result substitution is [ ].
3.17/2.04	
3.17/2.04	
3.17/2.04	
3.17/2.04	The rewrite sequence
3.17/2.04	
3.17/2.04	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.17/2.04	
3.17/2.04	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.17/2.04	
3.17/2.04	The pumping substitution is [X / n__from(X)].
3.17/2.04	
3.17/2.04	The result substitution is [ ].
3.17/2.04	
3.17/2.04	
3.17/2.04	
3.17/2.04	
3.17/2.04	----------------------------------------
3.17/2.04	
3.17/2.04	(10)
3.17/2.04	BOUNDS(EXP, INF)
3.43/2.08	EOF