3.52/1.96	WORST_CASE(NON_POLY, ?)
3.52/1.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.52/1.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.52/1.98	
3.52/1.98	
3.52/1.98	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.52/1.98	
3.52/1.98	(0) CpxTRS
3.52/1.98	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.52/1.98	(2) TRS for Loop Detection
3.52/1.98	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.52/1.98	(4) BEST
3.52/1.98	    (5) proven lower bound
3.52/1.98	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.52/1.98	        (7) BOUNDS(n^1, INF)
3.52/1.98	    (8) TRS for Loop Detection
3.52/1.98	        (9) DecreasingLoopProof [FINISHED, 34 ms]
3.52/1.98	        (10) BOUNDS(EXP, INF)
3.52/1.98	
3.52/1.98	
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(0)
3.52/1.98	Obligation:
3.52/1.98	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.52/1.98	
3.52/1.98	
3.52/1.98	The TRS R consists of the following rules:
3.52/1.98	
3.52/1.98	   from(X) -> cons(X, n__from(n__s(X)))
3.52/1.98	   sel(0, cons(X, XS)) -> X
3.52/1.98	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.52/1.98	   minus(X, 0) -> 0
3.52/1.98	   minus(s(X), s(Y)) -> minus(X, Y)
3.52/1.98	   quot(0, s(Y)) -> 0
3.52/1.98	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.52/1.98	   zWquot(XS, nil) -> nil
3.52/1.98	   zWquot(nil, XS) -> nil
3.52/1.98	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
3.52/1.98	   from(X) -> n__from(X)
3.52/1.98	   s(X) -> n__s(X)
3.52/1.98	   zWquot(X1, X2) -> n__zWquot(X1, X2)
3.52/1.98	   activate(n__from(X)) -> from(activate(X))
3.52/1.98	   activate(n__s(X)) -> s(activate(X))
3.52/1.98	   activate(n__zWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
3.52/1.98	   activate(X) -> X
3.52/1.98	
3.52/1.98	S is empty.
3.52/1.98	Rewrite Strategy: FULL
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.52/1.98	Transformed a relative TRS into a decreasing-loop problem.
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(2)
3.52/1.98	Obligation:
3.52/1.98	Analyzing the following TRS for decreasing loops:
3.52/1.98	
3.52/1.98	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.52/1.98	
3.52/1.98	
3.52/1.98	The TRS R consists of the following rules:
3.52/1.98	
3.52/1.98	   from(X) -> cons(X, n__from(n__s(X)))
3.52/1.98	   sel(0, cons(X, XS)) -> X
3.52/1.98	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.52/1.98	   minus(X, 0) -> 0
3.52/1.98	   minus(s(X), s(Y)) -> minus(X, Y)
3.52/1.98	   quot(0, s(Y)) -> 0
3.52/1.98	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.52/1.98	   zWquot(XS, nil) -> nil
3.52/1.98	   zWquot(nil, XS) -> nil
3.52/1.98	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
3.52/1.98	   from(X) -> n__from(X)
3.52/1.98	   s(X) -> n__s(X)
3.52/1.98	   zWquot(X1, X2) -> n__zWquot(X1, X2)
3.52/1.98	   activate(n__from(X)) -> from(activate(X))
3.52/1.98	   activate(n__s(X)) -> s(activate(X))
3.52/1.98	   activate(n__zWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
3.52/1.98	   activate(X) -> X
3.52/1.98	
3.52/1.98	S is empty.
3.52/1.98	Rewrite Strategy: FULL
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.52/1.98	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.52/1.98	
3.52/1.98	The rewrite sequence
3.52/1.98	
3.52/1.98	activate(n__zWquot(X1, X2)) ->^+ zWquot(activate(X1), activate(X2))
3.52/1.98	
3.52/1.98	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.52/1.98	
3.52/1.98	The pumping substitution is [X1 / n__zWquot(X1, X2)].
3.52/1.98	
3.52/1.98	The result substitution is [ ].
3.52/1.98	
3.52/1.98	
3.52/1.98	
3.52/1.98	
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(4)
3.52/1.98	Complex Obligation (BEST)
3.52/1.98	
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(5)
3.52/1.98	Obligation:
3.52/1.98	Proved the lower bound n^1 for the following obligation:
3.52/1.98	
3.52/1.98	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.52/1.98	
3.52/1.98	
3.52/1.98	The TRS R consists of the following rules:
3.52/1.98	
3.52/1.98	   from(X) -> cons(X, n__from(n__s(X)))
3.52/1.98	   sel(0, cons(X, XS)) -> X
3.52/1.98	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.52/1.98	   minus(X, 0) -> 0
3.52/1.98	   minus(s(X), s(Y)) -> minus(X, Y)
3.52/1.98	   quot(0, s(Y)) -> 0
3.52/1.98	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.52/1.98	   zWquot(XS, nil) -> nil
3.52/1.98	   zWquot(nil, XS) -> nil
3.52/1.98	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
3.52/1.98	   from(X) -> n__from(X)
3.52/1.98	   s(X) -> n__s(X)
3.52/1.98	   zWquot(X1, X2) -> n__zWquot(X1, X2)
3.52/1.98	   activate(n__from(X)) -> from(activate(X))
3.52/1.98	   activate(n__s(X)) -> s(activate(X))
3.52/1.98	   activate(n__zWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
3.52/1.98	   activate(X) -> X
3.52/1.98	
3.52/1.98	S is empty.
3.52/1.98	Rewrite Strategy: FULL
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(6) LowerBoundPropagationProof (FINISHED)
3.52/1.98	Propagated lower bound.
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(7)
3.52/1.98	BOUNDS(n^1, INF)
3.52/1.98	
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(8)
3.52/1.98	Obligation:
3.52/1.98	Analyzing the following TRS for decreasing loops:
3.52/1.98	
3.52/1.98	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.52/1.98	
3.52/1.98	
3.52/1.98	The TRS R consists of the following rules:
3.52/1.98	
3.52/1.98	   from(X) -> cons(X, n__from(n__s(X)))
3.52/1.98	   sel(0, cons(X, XS)) -> X
3.52/1.98	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
3.52/1.98	   minus(X, 0) -> 0
3.52/1.98	   minus(s(X), s(Y)) -> minus(X, Y)
3.52/1.98	   quot(0, s(Y)) -> 0
3.52/1.98	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
3.52/1.98	   zWquot(XS, nil) -> nil
3.52/1.98	   zWquot(nil, XS) -> nil
3.52/1.98	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
3.52/1.98	   from(X) -> n__from(X)
3.52/1.98	   s(X) -> n__s(X)
3.52/1.98	   zWquot(X1, X2) -> n__zWquot(X1, X2)
3.52/1.98	   activate(n__from(X)) -> from(activate(X))
3.52/1.98	   activate(n__s(X)) -> s(activate(X))
3.52/1.98	   activate(n__zWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
3.52/1.98	   activate(X) -> X
3.52/1.98	
3.52/1.98	S is empty.
3.52/1.98	Rewrite Strategy: FULL
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(9) DecreasingLoopProof (FINISHED)
3.52/1.98	The following loop(s) give(s) rise to the lower bound EXP:
3.52/1.98	
3.52/1.98	The rewrite sequence
3.52/1.98	
3.52/1.98	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.52/1.98	
3.52/1.98	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.52/1.98	
3.52/1.98	The pumping substitution is [X / n__from(X)].
3.52/1.98	
3.52/1.98	The result substitution is [ ].
3.52/1.98	
3.52/1.98	
3.52/1.98	
3.52/1.98	The rewrite sequence
3.52/1.98	
3.52/1.98	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.52/1.98	
3.52/1.98	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.52/1.98	
3.52/1.98	The pumping substitution is [X / n__from(X)].
3.52/1.98	
3.52/1.98	The result substitution is [ ].
3.52/1.98	
3.52/1.98	
3.52/1.98	
3.52/1.98	
3.52/1.98	----------------------------------------
3.52/1.98	
3.52/1.98	(10)
3.52/1.98	BOUNDS(EXP, INF)
3.64/2.03	EOF