3.75/1.77	WORST_CASE(NON_POLY, ?)
3.75/1.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.75/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.75/1.78	
3.75/1.78	
3.75/1.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.75/1.78	
3.75/1.78	(0) CpxTRS
3.75/1.78	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.75/1.78	(2) TRS for Loop Detection
3.75/1.78	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.75/1.78	(4) BEST
3.75/1.78	    (5) proven lower bound
3.75/1.78	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.75/1.78	        (7) BOUNDS(n^1, INF)
3.75/1.78	    (8) TRS for Loop Detection
3.75/1.78	        (9) DecreasingLoopProof [FINISHED, 105 ms]
3.75/1.78	        (10) BOUNDS(EXP, INF)
3.75/1.78	
3.75/1.78	
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(0)
3.75/1.78	Obligation:
3.75/1.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.75/1.78	
3.75/1.78	
3.75/1.78	The TRS R consists of the following rules:
3.75/1.78	
3.75/1.78	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.75/1.78	   sel(0, cons(X, Z)) -> X
3.75/1.78	   first(0, Z) -> nil
3.75/1.78	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.75/1.78	   from(X) -> cons(X, n__from(n__s(X)))
3.75/1.78	   sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z))
3.75/1.78	   sel1(0, cons(X, Z)) -> quote(X)
3.75/1.78	   first1(0, Z) -> nil1
3.75/1.78	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z)))
3.75/1.78	   quote(n__0) -> 01
3.75/1.78	   quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z)))
3.75/1.78	   quote1(n__nil) -> nil1
3.75/1.78	   quote(n__s(X)) -> s1(quote(activate(X)))
3.75/1.78	   quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z))
3.75/1.78	   quote1(n__first(X, Z)) -> first1(activate(X), activate(Z))
3.75/1.78	   unquote(01) -> 0
3.75/1.78	   unquote(s1(X)) -> s(unquote(X))
3.75/1.78	   unquote1(nil1) -> nil
3.75/1.78	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.75/1.78	   fcons(X, Z) -> cons(X, Z)
3.75/1.78	   first(X1, X2) -> n__first(X1, X2)
3.75/1.78	   from(X) -> n__from(X)
3.75/1.78	   s(X) -> n__s(X)
3.75/1.78	   0 -> n__0
3.75/1.78	   cons(X1, X2) -> n__cons(X1, X2)
3.75/1.78	   nil -> n__nil
3.75/1.78	   sel(X1, X2) -> n__sel(X1, X2)
3.75/1.78	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.75/1.78	   activate(n__from(X)) -> from(activate(X))
3.75/1.78	   activate(n__s(X)) -> s(activate(X))
3.75/1.78	   activate(n__0) -> 0
3.75/1.78	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.75/1.78	   activate(n__nil) -> nil
3.75/1.78	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.75/1.78	   activate(X) -> X
3.75/1.78	
3.75/1.78	S is empty.
3.75/1.78	Rewrite Strategy: FULL
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.75/1.78	Transformed a relative TRS into a decreasing-loop problem.
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(2)
3.75/1.78	Obligation:
3.75/1.78	Analyzing the following TRS for decreasing loops:
3.75/1.78	
3.75/1.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.75/1.78	
3.75/1.78	
3.75/1.78	The TRS R consists of the following rules:
3.75/1.78	
3.75/1.78	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.75/1.78	   sel(0, cons(X, Z)) -> X
3.75/1.78	   first(0, Z) -> nil
3.75/1.78	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.75/1.78	   from(X) -> cons(X, n__from(n__s(X)))
3.75/1.78	   sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z))
3.75/1.78	   sel1(0, cons(X, Z)) -> quote(X)
3.75/1.78	   first1(0, Z) -> nil1
3.75/1.78	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z)))
3.75/1.78	   quote(n__0) -> 01
3.75/1.78	   quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z)))
3.75/1.78	   quote1(n__nil) -> nil1
3.75/1.78	   quote(n__s(X)) -> s1(quote(activate(X)))
3.75/1.78	   quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z))
3.75/1.78	   quote1(n__first(X, Z)) -> first1(activate(X), activate(Z))
3.75/1.78	   unquote(01) -> 0
3.75/1.78	   unquote(s1(X)) -> s(unquote(X))
3.75/1.78	   unquote1(nil1) -> nil
3.75/1.78	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.75/1.78	   fcons(X, Z) -> cons(X, Z)
3.75/1.78	   first(X1, X2) -> n__first(X1, X2)
3.75/1.78	   from(X) -> n__from(X)
3.75/1.78	   s(X) -> n__s(X)
3.75/1.78	   0 -> n__0
3.75/1.78	   cons(X1, X2) -> n__cons(X1, X2)
3.75/1.78	   nil -> n__nil
3.75/1.78	   sel(X1, X2) -> n__sel(X1, X2)
3.75/1.78	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.75/1.78	   activate(n__from(X)) -> from(activate(X))
3.75/1.78	   activate(n__s(X)) -> s(activate(X))
3.75/1.78	   activate(n__0) -> 0
3.75/1.78	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.75/1.78	   activate(n__nil) -> nil
3.75/1.78	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.75/1.78	   activate(X) -> X
3.75/1.78	
3.75/1.78	S is empty.
3.75/1.78	Rewrite Strategy: FULL
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.75/1.78	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.75/1.78	
3.75/1.78	The rewrite sequence
3.75/1.78	
3.75/1.78	activate(n__s(X)) ->^+ s(activate(X))
3.75/1.78	
3.75/1.78	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.75/1.78	
3.75/1.78	The pumping substitution is [X / n__s(X)].
3.75/1.78	
3.75/1.78	The result substitution is [ ].
3.75/1.78	
3.75/1.78	
3.75/1.78	
3.75/1.78	
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(4)
3.75/1.78	Complex Obligation (BEST)
3.75/1.78	
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(5)
3.75/1.78	Obligation:
3.75/1.78	Proved the lower bound n^1 for the following obligation:
3.75/1.78	
3.75/1.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.75/1.78	
3.75/1.78	
3.75/1.78	The TRS R consists of the following rules:
3.75/1.78	
3.75/1.78	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.75/1.78	   sel(0, cons(X, Z)) -> X
3.75/1.78	   first(0, Z) -> nil
3.75/1.78	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.75/1.78	   from(X) -> cons(X, n__from(n__s(X)))
3.75/1.78	   sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z))
3.75/1.78	   sel1(0, cons(X, Z)) -> quote(X)
3.75/1.78	   first1(0, Z) -> nil1
3.75/1.78	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z)))
3.75/1.78	   quote(n__0) -> 01
3.75/1.78	   quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z)))
3.75/1.78	   quote1(n__nil) -> nil1
3.75/1.78	   quote(n__s(X)) -> s1(quote(activate(X)))
3.75/1.78	   quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z))
3.75/1.78	   quote1(n__first(X, Z)) -> first1(activate(X), activate(Z))
3.75/1.78	   unquote(01) -> 0
3.75/1.78	   unquote(s1(X)) -> s(unquote(X))
3.75/1.78	   unquote1(nil1) -> nil
3.75/1.78	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.75/1.78	   fcons(X, Z) -> cons(X, Z)
3.75/1.78	   first(X1, X2) -> n__first(X1, X2)
3.75/1.78	   from(X) -> n__from(X)
3.75/1.78	   s(X) -> n__s(X)
3.75/1.78	   0 -> n__0
3.75/1.78	   cons(X1, X2) -> n__cons(X1, X2)
3.75/1.78	   nil -> n__nil
3.75/1.78	   sel(X1, X2) -> n__sel(X1, X2)
3.75/1.78	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.75/1.78	   activate(n__from(X)) -> from(activate(X))
3.75/1.78	   activate(n__s(X)) -> s(activate(X))
3.75/1.78	   activate(n__0) -> 0
3.75/1.78	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
3.75/1.78	   activate(n__nil) -> nil
3.75/1.78	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.75/1.78	   activate(X) -> X
3.75/1.78	
3.75/1.78	S is empty.
3.75/1.78	Rewrite Strategy: FULL
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(6) LowerBoundPropagationProof (FINISHED)
3.75/1.78	Propagated lower bound.
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(7)
3.75/1.78	BOUNDS(n^1, INF)
3.75/1.78	
3.75/1.78	----------------------------------------
3.75/1.78	
3.75/1.78	(8)
3.75/1.78	Obligation:
3.75/1.78	Analyzing the following TRS for decreasing loops:
3.75/1.78	
3.75/1.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.75/1.78	
3.75/1.78	
3.75/1.78	The TRS R consists of the following rules:
3.75/1.78	
3.75/1.78	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.75/1.78	   sel(0, cons(X, Z)) -> X
3.75/1.78	   first(0, Z) -> nil
3.75/1.78	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.75/1.78	   from(X) -> cons(X, n__from(n__s(X)))
3.75/1.78	   sel1(s(X), cons(Y, Z)) -> sel1(X, activate(Z))
3.75/1.78	   sel1(0, cons(X, Z)) -> quote(X)
3.75/1.78	   first1(0, Z) -> nil1
3.75/1.78	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, activate(Z)))
3.75/1.78	   quote(n__0) -> 01
3.75/1.78	   quote1(n__cons(X, Z)) -> cons1(quote(activate(X)), quote1(activate(Z)))
3.75/1.78	   quote1(n__nil) -> nil1
3.75/1.78	   quote(n__s(X)) -> s1(quote(activate(X)))
3.75/1.78	   quote(n__sel(X, Z)) -> sel1(activate(X), activate(Z))
3.75/1.78	   quote1(n__first(X, Z)) -> first1(activate(X), activate(Z))
4.07/1.78	   unquote(01) -> 0
4.07/1.78	   unquote(s1(X)) -> s(unquote(X))
4.07/1.78	   unquote1(nil1) -> nil
4.07/1.78	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
4.07/1.78	   fcons(X, Z) -> cons(X, Z)
4.07/1.78	   first(X1, X2) -> n__first(X1, X2)
4.07/1.78	   from(X) -> n__from(X)
4.07/1.78	   s(X) -> n__s(X)
4.07/1.78	   0 -> n__0
4.07/1.78	   cons(X1, X2) -> n__cons(X1, X2)
4.07/1.78	   nil -> n__nil
4.07/1.78	   sel(X1, X2) -> n__sel(X1, X2)
4.07/1.78	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
4.07/1.78	   activate(n__from(X)) -> from(activate(X))
4.07/1.78	   activate(n__s(X)) -> s(activate(X))
4.07/1.78	   activate(n__0) -> 0
4.07/1.78	   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
4.07/1.78	   activate(n__nil) -> nil
4.07/1.78	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
4.07/1.78	   activate(X) -> X
4.07/1.78	
4.07/1.78	S is empty.
4.07/1.78	Rewrite Strategy: FULL
4.07/1.78	----------------------------------------
4.07/1.78	
4.07/1.78	(9) DecreasingLoopProof (FINISHED)
4.07/1.78	The following loop(s) give(s) rise to the lower bound EXP:
4.07/1.78	
4.07/1.78	The rewrite sequence
4.07/1.78	
4.07/1.78	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
4.07/1.78	
4.07/1.78	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.07/1.78	
4.07/1.78	The pumping substitution is [X / n__from(X)].
4.07/1.78	
4.07/1.78	The result substitution is [ ].
4.07/1.78	
4.07/1.78	
4.07/1.78	
4.07/1.78	The rewrite sequence
4.07/1.78	
4.07/1.78	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
4.07/1.78	
4.07/1.78	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
4.07/1.78	
4.07/1.78	The pumping substitution is [X / n__from(X)].
4.07/1.78	
4.07/1.78	The result substitution is [ ].
4.07/1.78	
4.07/1.78	
4.07/1.78	
4.07/1.78	
4.07/1.78	----------------------------------------
4.07/1.78	
4.07/1.78	(10)
4.07/1.78	BOUNDS(EXP, INF)
4.07/1.82	EOF