2.90/1.58	WORST_CASE(NON_POLY, ?)
2.90/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.90/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.90/1.59	
2.90/1.59	
2.90/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.59	
2.90/1.59	(0) CpxTRS
2.90/1.59	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.90/1.59	(2) TRS for Loop Detection
2.90/1.59	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.90/1.59	(4) BEST
2.90/1.59	    (5) proven lower bound
2.90/1.59	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
2.90/1.59	        (7) BOUNDS(n^1, INF)
2.90/1.59	    (8) TRS for Loop Detection
2.90/1.59	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
2.90/1.59	        (10) BOUNDS(INF, INF)
2.90/1.59	
2.90/1.59	
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(0)
2.90/1.59	Obligation:
2.90/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.59	
2.90/1.59	
2.90/1.59	The TRS R consists of the following rules:
2.90/1.59	
2.90/1.59	   from(X) -> cons(X, from(s(X)))
2.90/1.59	   2ndspos(0, Z) -> rnil
2.90/1.59	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
2.90/1.59	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
2.90/1.59	   2ndsneg(0, Z) -> rnil
2.90/1.59	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
2.90/1.59	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
2.90/1.59	   pi(X) -> 2ndspos(X, from(0))
2.90/1.59	   plus(0, Y) -> Y
2.90/1.59	   plus(s(X), Y) -> s(plus(X, Y))
2.90/1.59	   times(0, Y) -> 0
2.90/1.59	   times(s(X), Y) -> plus(Y, times(X, Y))
2.90/1.59	   square(X) -> times(X, X)
2.90/1.59	
2.90/1.59	S is empty.
2.90/1.59	Rewrite Strategy: INNERMOST
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.90/1.59	Transformed a relative TRS into a decreasing-loop problem.
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(2)
2.90/1.59	Obligation:
2.90/1.59	Analyzing the following TRS for decreasing loops:
2.90/1.59	
2.90/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.59	
2.90/1.59	
2.90/1.59	The TRS R consists of the following rules:
2.90/1.59	
2.90/1.59	   from(X) -> cons(X, from(s(X)))
2.90/1.59	   2ndspos(0, Z) -> rnil
2.90/1.59	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
2.90/1.59	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
2.90/1.59	   2ndsneg(0, Z) -> rnil
2.90/1.59	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
2.90/1.59	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
2.90/1.59	   pi(X) -> 2ndspos(X, from(0))
2.90/1.59	   plus(0, Y) -> Y
2.90/1.59	   plus(s(X), Y) -> s(plus(X, Y))
2.90/1.59	   times(0, Y) -> 0
2.90/1.59	   times(s(X), Y) -> plus(Y, times(X, Y))
2.90/1.59	   square(X) -> times(X, X)
2.90/1.59	
2.90/1.59	S is empty.
2.90/1.59	Rewrite Strategy: INNERMOST
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(3) DecreasingLoopProof (LOWER BOUND(ID))
2.90/1.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.90/1.59	
2.90/1.59	The rewrite sequence
2.90/1.59	
2.90/1.59	plus(s(X), Y) ->^+ s(plus(X, Y))
2.90/1.59	
2.90/1.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
2.90/1.59	
2.90/1.59	The pumping substitution is [X / s(X)].
2.90/1.59	
2.90/1.59	The result substitution is [ ].
2.90/1.59	
2.90/1.59	
2.90/1.59	
2.90/1.59	
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(4)
2.90/1.59	Complex Obligation (BEST)
2.90/1.59	
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(5)
2.90/1.59	Obligation:
2.90/1.59	Proved the lower bound n^1 for the following obligation:
2.90/1.59	
2.90/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.59	
2.90/1.59	
2.90/1.59	The TRS R consists of the following rules:
2.90/1.59	
2.90/1.59	   from(X) -> cons(X, from(s(X)))
2.90/1.59	   2ndspos(0, Z) -> rnil
2.90/1.59	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
2.90/1.59	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
2.90/1.59	   2ndsneg(0, Z) -> rnil
2.90/1.59	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
2.90/1.59	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
2.90/1.59	   pi(X) -> 2ndspos(X, from(0))
2.90/1.59	   plus(0, Y) -> Y
2.90/1.59	   plus(s(X), Y) -> s(plus(X, Y))
2.90/1.59	   times(0, Y) -> 0
2.90/1.59	   times(s(X), Y) -> plus(Y, times(X, Y))
2.90/1.59	   square(X) -> times(X, X)
2.90/1.59	
2.90/1.59	S is empty.
2.90/1.59	Rewrite Strategy: INNERMOST
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(6) LowerBoundPropagationProof (FINISHED)
2.90/1.59	Propagated lower bound.
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(7)
2.90/1.59	BOUNDS(n^1, INF)
2.90/1.59	
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(8)
2.90/1.59	Obligation:
2.90/1.59	Analyzing the following TRS for decreasing loops:
2.90/1.59	
2.90/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.59	
2.90/1.59	
2.90/1.59	The TRS R consists of the following rules:
2.90/1.59	
2.90/1.59	   from(X) -> cons(X, from(s(X)))
2.90/1.59	   2ndspos(0, Z) -> rnil
2.90/1.59	   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
2.90/1.59	   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
2.90/1.59	   2ndsneg(0, Z) -> rnil
2.90/1.59	   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
2.90/1.59	   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
2.90/1.59	   pi(X) -> 2ndspos(X, from(0))
2.90/1.59	   plus(0, Y) -> Y
2.90/1.59	   plus(s(X), Y) -> s(plus(X, Y))
2.90/1.59	   times(0, Y) -> 0
2.90/1.59	   times(s(X), Y) -> plus(Y, times(X, Y))
2.90/1.59	   square(X) -> times(X, X)
2.90/1.59	
2.90/1.59	S is empty.
2.90/1.59	Rewrite Strategy: INNERMOST
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(9) InfiniteLowerBoundProof (FINISHED)
2.90/1.59	The following loop proves infinite runtime complexity:
2.90/1.59	
2.90/1.59	The rewrite sequence
2.90/1.59	
2.90/1.59	from(X) ->^+ cons(X, from(s(X)))
2.90/1.59	
2.90/1.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
2.90/1.59	
2.90/1.59	The pumping substitution is [ ].
2.90/1.59	
2.90/1.59	The result substitution is [X / s(X)].
2.90/1.59	
2.90/1.59	
2.90/1.59	
2.90/1.59	
2.90/1.59	----------------------------------------
2.90/1.59	
2.90/1.59	(10)
2.90/1.59	BOUNDS(INF, INF)
3.19/1.62	EOF