3.93/1.78	WORST_CASE(NON_POLY, ?)
3.93/1.80	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.93/1.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.93/1.80	
3.93/1.80	
3.93/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.93/1.80	
3.93/1.80	(0) CpxTRS
3.93/1.80	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.93/1.80	(2) TRS for Loop Detection
3.93/1.80	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.93/1.80	(4) BEST
3.93/1.80	    (5) proven lower bound
3.93/1.80	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.93/1.80	        (7) BOUNDS(n^1, INF)
3.93/1.80	    (8) TRS for Loop Detection
3.93/1.80	        (9) DecreasingLoopProof [FINISHED, 81 ms]
3.93/1.80	        (10) BOUNDS(EXP, INF)
3.93/1.80	
3.93/1.80	
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(0)
3.93/1.80	Obligation:
3.93/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.93/1.80	
3.93/1.80	
3.93/1.80	The TRS R consists of the following rules:
3.93/1.80	
3.93/1.80	   a__from(X) -> cons(mark(X), from(s(X)))
3.93/1.80	   a__2ndspos(0, Z) -> rnil
3.93/1.80	   a__2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
3.93/1.80	   a__2ndsneg(0, Z) -> rnil
3.93/1.80	   a__2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
3.93/1.80	   a__pi(X) -> a__2ndspos(mark(X), a__from(0))
3.93/1.80	   a__plus(0, Y) -> mark(Y)
3.93/1.80	   a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y)))
3.93/1.80	   a__times(0, Y) -> 0
3.93/1.80	   a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y)))
3.93/1.80	   a__square(X) -> a__times(mark(X), mark(X))
3.93/1.80	   mark(from(X)) -> a__from(mark(X))
3.93/1.80	   mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2))
3.93/1.80	   mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2))
3.93/1.80	   mark(pi(X)) -> a__pi(mark(X))
3.93/1.80	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.93/1.80	   mark(times(X1, X2)) -> a__times(mark(X1), mark(X2))
3.93/1.80	   mark(square(X)) -> a__square(mark(X))
3.93/1.80	   mark(0) -> 0
3.93/1.80	   mark(s(X)) -> s(mark(X))
3.93/1.80	   mark(posrecip(X)) -> posrecip(mark(X))
3.93/1.80	   mark(negrecip(X)) -> negrecip(mark(X))
3.93/1.80	   mark(nil) -> nil
3.93/1.80	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.93/1.80	   mark(rnil) -> rnil
3.93/1.80	   mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2))
3.93/1.80	   a__from(X) -> from(X)
3.93/1.80	   a__2ndspos(X1, X2) -> 2ndspos(X1, X2)
3.93/1.80	   a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2)
3.93/1.80	   a__pi(X) -> pi(X)
3.93/1.80	   a__plus(X1, X2) -> plus(X1, X2)
3.93/1.80	   a__times(X1, X2) -> times(X1, X2)
3.93/1.80	   a__square(X) -> square(X)
3.93/1.80	
3.93/1.80	S is empty.
3.93/1.80	Rewrite Strategy: FULL
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.93/1.80	Transformed a relative TRS into a decreasing-loop problem.
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(2)
3.93/1.80	Obligation:
3.93/1.80	Analyzing the following TRS for decreasing loops:
3.93/1.80	
3.93/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.93/1.80	
3.93/1.80	
3.93/1.80	The TRS R consists of the following rules:
3.93/1.80	
3.93/1.80	   a__from(X) -> cons(mark(X), from(s(X)))
3.93/1.80	   a__2ndspos(0, Z) -> rnil
3.93/1.80	   a__2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
3.93/1.80	   a__2ndsneg(0, Z) -> rnil
3.93/1.80	   a__2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
3.93/1.80	   a__pi(X) -> a__2ndspos(mark(X), a__from(0))
3.93/1.80	   a__plus(0, Y) -> mark(Y)
3.93/1.80	   a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y)))
3.93/1.80	   a__times(0, Y) -> 0
3.93/1.80	   a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y)))
3.93/1.80	   a__square(X) -> a__times(mark(X), mark(X))
3.93/1.80	   mark(from(X)) -> a__from(mark(X))
3.93/1.80	   mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2))
3.93/1.80	   mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2))
3.93/1.80	   mark(pi(X)) -> a__pi(mark(X))
3.93/1.80	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.93/1.80	   mark(times(X1, X2)) -> a__times(mark(X1), mark(X2))
3.93/1.80	   mark(square(X)) -> a__square(mark(X))
3.93/1.80	   mark(0) -> 0
3.93/1.80	   mark(s(X)) -> s(mark(X))
3.93/1.80	   mark(posrecip(X)) -> posrecip(mark(X))
3.93/1.80	   mark(negrecip(X)) -> negrecip(mark(X))
3.93/1.80	   mark(nil) -> nil
3.93/1.80	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.93/1.80	   mark(rnil) -> rnil
3.93/1.80	   mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2))
3.93/1.80	   a__from(X) -> from(X)
3.93/1.80	   a__2ndspos(X1, X2) -> 2ndspos(X1, X2)
3.93/1.80	   a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2)
3.93/1.80	   a__pi(X) -> pi(X)
3.93/1.80	   a__plus(X1, X2) -> plus(X1, X2)
3.93/1.80	   a__times(X1, X2) -> times(X1, X2)
3.93/1.80	   a__square(X) -> square(X)
3.93/1.80	
3.93/1.80	S is empty.
3.93/1.80	Rewrite Strategy: FULL
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.93/1.80	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.93/1.80	
3.93/1.80	The rewrite sequence
3.93/1.80	
3.93/1.80	mark(2ndsneg(X1, X2)) ->^+ a__2ndsneg(mark(X1), mark(X2))
3.93/1.80	
3.93/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.93/1.80	
3.93/1.80	The pumping substitution is [X1 / 2ndsneg(X1, X2)].
3.93/1.80	
3.93/1.80	The result substitution is [ ].
3.93/1.80	
3.93/1.80	
3.93/1.80	
3.93/1.80	
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(4)
3.93/1.80	Complex Obligation (BEST)
3.93/1.80	
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(5)
3.93/1.80	Obligation:
3.93/1.80	Proved the lower bound n^1 for the following obligation:
3.93/1.80	
3.93/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.93/1.80	
3.93/1.80	
3.93/1.80	The TRS R consists of the following rules:
3.93/1.80	
3.93/1.80	   a__from(X) -> cons(mark(X), from(s(X)))
3.93/1.80	   a__2ndspos(0, Z) -> rnil
3.93/1.80	   a__2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
3.93/1.80	   a__2ndsneg(0, Z) -> rnil
3.93/1.80	   a__2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
3.93/1.80	   a__pi(X) -> a__2ndspos(mark(X), a__from(0))
3.93/1.80	   a__plus(0, Y) -> mark(Y)
3.93/1.80	   a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y)))
3.93/1.80	   a__times(0, Y) -> 0
3.93/1.80	   a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y)))
3.93/1.80	   a__square(X) -> a__times(mark(X), mark(X))
3.93/1.80	   mark(from(X)) -> a__from(mark(X))
3.93/1.80	   mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2))
3.93/1.80	   mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2))
3.93/1.80	   mark(pi(X)) -> a__pi(mark(X))
3.93/1.80	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.93/1.80	   mark(times(X1, X2)) -> a__times(mark(X1), mark(X2))
3.93/1.80	   mark(square(X)) -> a__square(mark(X))
3.93/1.80	   mark(0) -> 0
3.93/1.80	   mark(s(X)) -> s(mark(X))
3.93/1.80	   mark(posrecip(X)) -> posrecip(mark(X))
3.93/1.80	   mark(negrecip(X)) -> negrecip(mark(X))
3.93/1.80	   mark(nil) -> nil
3.93/1.80	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.93/1.80	   mark(rnil) -> rnil
3.93/1.80	   mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2))
3.93/1.80	   a__from(X) -> from(X)
3.93/1.80	   a__2ndspos(X1, X2) -> 2ndspos(X1, X2)
3.93/1.80	   a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2)
3.93/1.80	   a__pi(X) -> pi(X)
3.93/1.80	   a__plus(X1, X2) -> plus(X1, X2)
3.93/1.80	   a__times(X1, X2) -> times(X1, X2)
3.93/1.80	   a__square(X) -> square(X)
3.93/1.80	
3.93/1.80	S is empty.
3.93/1.80	Rewrite Strategy: FULL
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(6) LowerBoundPropagationProof (FINISHED)
3.93/1.80	Propagated lower bound.
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(7)
3.93/1.80	BOUNDS(n^1, INF)
3.93/1.80	
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(8)
3.93/1.80	Obligation:
3.93/1.80	Analyzing the following TRS for decreasing loops:
3.93/1.80	
3.93/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.93/1.80	
3.93/1.80	
3.93/1.80	The TRS R consists of the following rules:
3.93/1.80	
3.93/1.80	   a__from(X) -> cons(mark(X), from(s(X)))
3.93/1.80	   a__2ndspos(0, Z) -> rnil
3.93/1.80	   a__2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
3.93/1.80	   a__2ndsneg(0, Z) -> rnil
3.93/1.80	   a__2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
3.93/1.80	   a__pi(X) -> a__2ndspos(mark(X), a__from(0))
3.93/1.80	   a__plus(0, Y) -> mark(Y)
3.93/1.80	   a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y)))
3.93/1.80	   a__times(0, Y) -> 0
3.93/1.80	   a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y)))
3.93/1.80	   a__square(X) -> a__times(mark(X), mark(X))
3.93/1.80	   mark(from(X)) -> a__from(mark(X))
3.93/1.80	   mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2))
3.93/1.80	   mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2))
3.93/1.80	   mark(pi(X)) -> a__pi(mark(X))
3.93/1.80	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.93/1.80	   mark(times(X1, X2)) -> a__times(mark(X1), mark(X2))
3.93/1.80	   mark(square(X)) -> a__square(mark(X))
3.93/1.80	   mark(0) -> 0
3.93/1.80	   mark(s(X)) -> s(mark(X))
3.93/1.80	   mark(posrecip(X)) -> posrecip(mark(X))
3.93/1.80	   mark(negrecip(X)) -> negrecip(mark(X))
3.93/1.80	   mark(nil) -> nil
3.93/1.80	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.93/1.80	   mark(rnil) -> rnil
3.93/1.80	   mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2))
3.93/1.80	   a__from(X) -> from(X)
3.93/1.80	   a__2ndspos(X1, X2) -> 2ndspos(X1, X2)
3.93/1.80	   a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2)
3.93/1.80	   a__pi(X) -> pi(X)
3.93/1.80	   a__plus(X1, X2) -> plus(X1, X2)
3.93/1.80	   a__times(X1, X2) -> times(X1, X2)
3.93/1.80	   a__square(X) -> square(X)
3.93/1.80	
3.93/1.80	S is empty.
3.93/1.80	Rewrite Strategy: FULL
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(9) DecreasingLoopProof (FINISHED)
3.93/1.80	The following loop(s) give(s) rise to the lower bound EXP:
3.93/1.80	
3.93/1.80	The rewrite sequence
3.93/1.80	
3.93/1.80	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.93/1.80	
3.93/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.93/1.80	
3.93/1.80	The pumping substitution is [X / from(X)].
3.93/1.80	
3.93/1.80	The result substitution is [ ].
3.93/1.80	
3.93/1.80	
3.93/1.80	
3.93/1.80	The rewrite sequence
3.93/1.80	
3.93/1.80	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.93/1.80	
3.93/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.93/1.80	
3.93/1.80	The pumping substitution is [X / from(X)].
3.93/1.80	
3.93/1.80	The result substitution is [ ].
3.93/1.80	
3.93/1.80	
3.93/1.80	
3.93/1.80	
3.93/1.80	----------------------------------------
3.93/1.80	
3.93/1.80	(10)
3.93/1.80	BOUNDS(EXP, INF)
4.16/1.83	EOF