3.78/1.78	WORST_CASE(NON_POLY, ?)
3.78/1.80	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.78/1.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.78/1.80	
3.78/1.80	
3.78/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.80	
3.78/1.80	(0) CpxTRS
3.78/1.80	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.78/1.80	(2) TRS for Loop Detection
3.78/1.80	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.78/1.80	(4) BEST
3.78/1.80	    (5) proven lower bound
3.78/1.80	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.78/1.80	        (7) BOUNDS(n^1, INF)
3.78/1.80	    (8) TRS for Loop Detection
3.78/1.80	        (9) DecreasingLoopProof [FINISHED, 17 ms]
3.78/1.80	        (10) BOUNDS(EXP, INF)
3.78/1.80	
3.78/1.80	
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(0)
3.78/1.80	Obligation:
3.78/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.80	
3.78/1.80	
3.78/1.80	The TRS R consists of the following rules:
3.78/1.80	
3.78/1.80	   dbl(0) -> 0
3.78/1.80	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.78/1.80	   dbls(nil) -> nil
3.78/1.80	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.78/1.80	   sel(0, cons(X, Y)) -> activate(X)
3.78/1.80	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.78/1.80	   indx(nil, X) -> nil
3.78/1.80	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.78/1.80	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.78/1.80	   s(X) -> n__s(X)
3.78/1.80	   dbl(X) -> n__dbl(X)
3.78/1.80	   dbls(X) -> n__dbls(X)
3.78/1.80	   sel(X1, X2) -> n__sel(X1, X2)
3.78/1.80	   indx(X1, X2) -> n__indx(X1, X2)
3.78/1.80	   from(X) -> n__from(X)
3.78/1.80	   activate(n__s(X)) -> s(X)
3.78/1.80	   activate(n__dbl(X)) -> dbl(activate(X))
3.78/1.80	   activate(n__dbls(X)) -> dbls(activate(X))
3.78/1.80	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.78/1.80	   activate(n__indx(X1, X2)) -> indx(activate(X1), X2)
3.78/1.80	   activate(n__from(X)) -> from(X)
3.78/1.80	   activate(X) -> X
3.78/1.80	
3.78/1.80	S is empty.
3.78/1.80	Rewrite Strategy: FULL
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.78/1.80	Transformed a relative TRS into a decreasing-loop problem.
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(2)
3.78/1.80	Obligation:
3.78/1.80	Analyzing the following TRS for decreasing loops:
3.78/1.80	
3.78/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.80	
3.78/1.80	
3.78/1.80	The TRS R consists of the following rules:
3.78/1.80	
3.78/1.80	   dbl(0) -> 0
3.78/1.80	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.78/1.80	   dbls(nil) -> nil
3.78/1.80	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.78/1.80	   sel(0, cons(X, Y)) -> activate(X)
3.78/1.80	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.78/1.80	   indx(nil, X) -> nil
3.78/1.80	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.78/1.80	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.78/1.80	   s(X) -> n__s(X)
3.78/1.80	   dbl(X) -> n__dbl(X)
3.78/1.80	   dbls(X) -> n__dbls(X)
3.78/1.80	   sel(X1, X2) -> n__sel(X1, X2)
3.78/1.80	   indx(X1, X2) -> n__indx(X1, X2)
3.78/1.80	   from(X) -> n__from(X)
3.78/1.80	   activate(n__s(X)) -> s(X)
3.78/1.80	   activate(n__dbl(X)) -> dbl(activate(X))
3.78/1.80	   activate(n__dbls(X)) -> dbls(activate(X))
3.78/1.80	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.78/1.80	   activate(n__indx(X1, X2)) -> indx(activate(X1), X2)
3.78/1.80	   activate(n__from(X)) -> from(X)
3.78/1.80	   activate(X) -> X
3.78/1.80	
3.78/1.80	S is empty.
3.78/1.80	Rewrite Strategy: FULL
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.78/1.80	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.78/1.80	
3.78/1.80	The rewrite sequence
3.78/1.80	
3.78/1.80	activate(n__dbl(X)) ->^+ dbl(activate(X))
3.78/1.80	
3.78/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.78/1.80	
3.78/1.80	The pumping substitution is [X / n__dbl(X)].
3.78/1.80	
3.78/1.80	The result substitution is [ ].
3.78/1.80	
3.78/1.80	
3.78/1.80	
3.78/1.80	
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(4)
3.78/1.80	Complex Obligation (BEST)
3.78/1.80	
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(5)
3.78/1.80	Obligation:
3.78/1.80	Proved the lower bound n^1 for the following obligation:
3.78/1.80	
3.78/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.80	
3.78/1.80	
3.78/1.80	The TRS R consists of the following rules:
3.78/1.80	
3.78/1.80	   dbl(0) -> 0
3.78/1.80	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.78/1.80	   dbls(nil) -> nil
3.78/1.80	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.78/1.80	   sel(0, cons(X, Y)) -> activate(X)
3.78/1.80	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.78/1.80	   indx(nil, X) -> nil
3.78/1.80	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.78/1.80	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.78/1.80	   s(X) -> n__s(X)
3.78/1.80	   dbl(X) -> n__dbl(X)
3.78/1.80	   dbls(X) -> n__dbls(X)
3.78/1.80	   sel(X1, X2) -> n__sel(X1, X2)
3.78/1.80	   indx(X1, X2) -> n__indx(X1, X2)
3.78/1.80	   from(X) -> n__from(X)
3.78/1.80	   activate(n__s(X)) -> s(X)
3.78/1.80	   activate(n__dbl(X)) -> dbl(activate(X))
3.78/1.80	   activate(n__dbls(X)) -> dbls(activate(X))
3.78/1.80	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.78/1.80	   activate(n__indx(X1, X2)) -> indx(activate(X1), X2)
3.78/1.80	   activate(n__from(X)) -> from(X)
3.78/1.80	   activate(X) -> X
3.78/1.80	
3.78/1.80	S is empty.
3.78/1.80	Rewrite Strategy: FULL
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(6) LowerBoundPropagationProof (FINISHED)
3.78/1.80	Propagated lower bound.
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(7)
3.78/1.80	BOUNDS(n^1, INF)
3.78/1.80	
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(8)
3.78/1.80	Obligation:
3.78/1.80	Analyzing the following TRS for decreasing loops:
3.78/1.80	
3.78/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.78/1.80	
3.78/1.80	
3.78/1.80	The TRS R consists of the following rules:
3.78/1.80	
3.78/1.80	   dbl(0) -> 0
3.78/1.80	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.78/1.80	   dbls(nil) -> nil
3.78/1.80	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.78/1.80	   sel(0, cons(X, Y)) -> activate(X)
3.78/1.80	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.78/1.80	   indx(nil, X) -> nil
3.78/1.80	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.78/1.80	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.78/1.80	   s(X) -> n__s(X)
3.78/1.80	   dbl(X) -> n__dbl(X)
3.78/1.80	   dbls(X) -> n__dbls(X)
3.78/1.80	   sel(X1, X2) -> n__sel(X1, X2)
3.78/1.80	   indx(X1, X2) -> n__indx(X1, X2)
3.78/1.80	   from(X) -> n__from(X)
3.78/1.80	   activate(n__s(X)) -> s(X)
3.78/1.80	   activate(n__dbl(X)) -> dbl(activate(X))
3.78/1.80	   activate(n__dbls(X)) -> dbls(activate(X))
3.78/1.80	   activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
3.78/1.80	   activate(n__indx(X1, X2)) -> indx(activate(X1), X2)
3.78/1.80	   activate(n__from(X)) -> from(X)
3.78/1.80	   activate(X) -> X
3.78/1.80	
3.78/1.80	S is empty.
3.78/1.80	Rewrite Strategy: FULL
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(9) DecreasingLoopProof (FINISHED)
3.78/1.80	The following loop(s) give(s) rise to the lower bound EXP:
3.78/1.80	
3.78/1.80	The rewrite sequence
3.78/1.80	
3.78/1.80	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.78/1.80	
3.78/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.78/1.80	
3.78/1.80	The pumping substitution is [X / n__from(X)].
3.78/1.80	
3.78/1.80	The result substitution is [ ].
3.78/1.80	
3.78/1.80	
3.78/1.80	
3.78/1.80	The rewrite sequence
3.78/1.80	
3.78/1.80	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.78/1.80	
3.78/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.78/1.80	
3.78/1.80	The pumping substitution is [X / n__from(X)].
3.78/1.80	
3.78/1.80	The result substitution is [ ].
3.78/1.80	
3.78/1.80	
3.78/1.80	
3.78/1.80	
3.78/1.80	----------------------------------------
3.78/1.80	
3.78/1.80	(10)
3.78/1.80	BOUNDS(EXP, INF)
3.87/1.90	EOF