3.53/1.66	WORST_CASE(NON_POLY, ?)
3.53/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.53/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.53/1.67	
3.53/1.67	
3.53/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.53/1.67	
3.53/1.67	(0) CpxTRS
3.53/1.67	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.53/1.67	(2) TRS for Loop Detection
3.53/1.67	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.53/1.67	(4) BEST
3.53/1.67	    (5) proven lower bound
3.53/1.67	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.53/1.67	        (7) BOUNDS(n^1, INF)
3.53/1.67	    (8) TRS for Loop Detection
3.53/1.67	        (9) DecreasingLoopProof [FINISHED, 39 ms]
3.53/1.67	        (10) BOUNDS(EXP, INF)
3.53/1.67	
3.53/1.67	
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(0)
3.53/1.67	Obligation:
3.53/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.53/1.67	
3.53/1.67	
3.53/1.67	The TRS R consists of the following rules:
3.53/1.67	
3.53/1.67	   app(nil, YS) -> YS
3.53/1.67	   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
3.53/1.67	   from(X) -> cons(X, n__from(n__s(X)))
3.53/1.67	   zWadr(nil, YS) -> nil
3.53/1.67	   zWadr(XS, nil) -> nil
3.53/1.67	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
3.53/1.67	   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
3.53/1.67	   app(X1, X2) -> n__app(X1, X2)
3.53/1.67	   from(X) -> n__from(X)
3.53/1.67	   s(X) -> n__s(X)
3.53/1.67	   nil -> n__nil
3.53/1.67	   zWadr(X1, X2) -> n__zWadr(X1, X2)
3.53/1.67	   prefix(X) -> n__prefix(X)
3.53/1.67	   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
3.53/1.67	   activate(n__from(X)) -> from(activate(X))
3.53/1.67	   activate(n__s(X)) -> s(activate(X))
3.53/1.67	   activate(n__nil) -> nil
3.53/1.67	   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
3.53/1.67	   activate(n__prefix(X)) -> prefix(activate(X))
3.53/1.67	   activate(X) -> X
3.53/1.67	
3.53/1.67	S is empty.
3.53/1.67	Rewrite Strategy: FULL
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.53/1.67	Transformed a relative TRS into a decreasing-loop problem.
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(2)
3.53/1.67	Obligation:
3.53/1.67	Analyzing the following TRS for decreasing loops:
3.53/1.67	
3.53/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.53/1.67	
3.53/1.67	
3.53/1.67	The TRS R consists of the following rules:
3.53/1.67	
3.53/1.67	   app(nil, YS) -> YS
3.53/1.67	   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
3.53/1.67	   from(X) -> cons(X, n__from(n__s(X)))
3.53/1.67	   zWadr(nil, YS) -> nil
3.53/1.67	   zWadr(XS, nil) -> nil
3.53/1.67	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
3.53/1.67	   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
3.53/1.67	   app(X1, X2) -> n__app(X1, X2)
3.53/1.67	   from(X) -> n__from(X)
3.53/1.67	   s(X) -> n__s(X)
3.53/1.67	   nil -> n__nil
3.53/1.67	   zWadr(X1, X2) -> n__zWadr(X1, X2)
3.53/1.67	   prefix(X) -> n__prefix(X)
3.53/1.67	   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
3.53/1.67	   activate(n__from(X)) -> from(activate(X))
3.53/1.67	   activate(n__s(X)) -> s(activate(X))
3.53/1.67	   activate(n__nil) -> nil
3.53/1.67	   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
3.53/1.67	   activate(n__prefix(X)) -> prefix(activate(X))
3.53/1.67	   activate(X) -> X
3.53/1.67	
3.53/1.67	S is empty.
3.53/1.67	Rewrite Strategy: FULL
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.53/1.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.53/1.67	
3.53/1.67	The rewrite sequence
3.53/1.67	
3.53/1.67	activate(n__s(X)) ->^+ s(activate(X))
3.53/1.67	
3.53/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.53/1.67	
3.53/1.67	The pumping substitution is [X / n__s(X)].
3.53/1.67	
3.53/1.67	The result substitution is [ ].
3.53/1.67	
3.53/1.67	
3.53/1.67	
3.53/1.67	
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(4)
3.53/1.67	Complex Obligation (BEST)
3.53/1.67	
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(5)
3.53/1.67	Obligation:
3.53/1.67	Proved the lower bound n^1 for the following obligation:
3.53/1.67	
3.53/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.53/1.67	
3.53/1.67	
3.53/1.67	The TRS R consists of the following rules:
3.53/1.67	
3.53/1.67	   app(nil, YS) -> YS
3.53/1.67	   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
3.53/1.67	   from(X) -> cons(X, n__from(n__s(X)))
3.53/1.67	   zWadr(nil, YS) -> nil
3.53/1.67	   zWadr(XS, nil) -> nil
3.53/1.67	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
3.53/1.67	   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
3.53/1.67	   app(X1, X2) -> n__app(X1, X2)
3.53/1.67	   from(X) -> n__from(X)
3.53/1.67	   s(X) -> n__s(X)
3.53/1.67	   nil -> n__nil
3.53/1.67	   zWadr(X1, X2) -> n__zWadr(X1, X2)
3.53/1.67	   prefix(X) -> n__prefix(X)
3.53/1.67	   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
3.53/1.67	   activate(n__from(X)) -> from(activate(X))
3.53/1.67	   activate(n__s(X)) -> s(activate(X))
3.53/1.67	   activate(n__nil) -> nil
3.53/1.67	   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
3.53/1.67	   activate(n__prefix(X)) -> prefix(activate(X))
3.53/1.67	   activate(X) -> X
3.53/1.67	
3.53/1.67	S is empty.
3.53/1.67	Rewrite Strategy: FULL
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(6) LowerBoundPropagationProof (FINISHED)
3.53/1.67	Propagated lower bound.
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(7)
3.53/1.67	BOUNDS(n^1, INF)
3.53/1.67	
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(8)
3.53/1.67	Obligation:
3.53/1.67	Analyzing the following TRS for decreasing loops:
3.53/1.67	
3.53/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.53/1.67	
3.53/1.67	
3.53/1.67	The TRS R consists of the following rules:
3.53/1.67	
3.53/1.67	   app(nil, YS) -> YS
3.53/1.67	   app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS))
3.53/1.67	   from(X) -> cons(X, n__from(n__s(X)))
3.53/1.67	   zWadr(nil, YS) -> nil
3.53/1.67	   zWadr(XS, nil) -> nil
3.53/1.67	   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
3.53/1.67	   prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L)))
3.53/1.67	   app(X1, X2) -> n__app(X1, X2)
3.53/1.67	   from(X) -> n__from(X)
3.53/1.67	   s(X) -> n__s(X)
3.53/1.67	   nil -> n__nil
3.53/1.67	   zWadr(X1, X2) -> n__zWadr(X1, X2)
3.53/1.67	   prefix(X) -> n__prefix(X)
3.53/1.67	   activate(n__app(X1, X2)) -> app(activate(X1), activate(X2))
3.53/1.67	   activate(n__from(X)) -> from(activate(X))
3.53/1.67	   activate(n__s(X)) -> s(activate(X))
3.53/1.67	   activate(n__nil) -> nil
3.53/1.67	   activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
3.53/1.67	   activate(n__prefix(X)) -> prefix(activate(X))
3.53/1.67	   activate(X) -> X
3.53/1.67	
3.53/1.67	S is empty.
3.53/1.67	Rewrite Strategy: FULL
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(9) DecreasingLoopProof (FINISHED)
3.53/1.67	The following loop(s) give(s) rise to the lower bound EXP:
3.53/1.67	
3.53/1.67	The rewrite sequence
3.53/1.67	
3.53/1.67	activate(n__prefix(X)) ->^+ cons(nil, n__zWadr(activate(X), n__prefix(activate(X))))
3.53/1.67	
3.53/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
3.53/1.67	
3.53/1.67	The pumping substitution is [X / n__prefix(X)].
3.53/1.67	
3.53/1.67	The result substitution is [ ].
3.53/1.67	
3.53/1.67	
3.53/1.67	
3.53/1.67	The rewrite sequence
3.53/1.67	
3.53/1.67	activate(n__prefix(X)) ->^+ cons(nil, n__zWadr(activate(X), n__prefix(activate(X))))
3.53/1.67	
3.53/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,0].
3.53/1.67	
3.53/1.67	The pumping substitution is [X / n__prefix(X)].
3.53/1.67	
3.53/1.67	The result substitution is [ ].
3.53/1.67	
3.53/1.67	
3.53/1.67	
3.53/1.67	
3.53/1.67	----------------------------------------
3.53/1.67	
3.53/1.67	(10)
3.53/1.67	BOUNDS(EXP, INF)
3.53/1.70	EOF