3.41/1.75	WORST_CASE(NON_POLY, ?)
3.41/1.76	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.76	
3.41/1.76	
3.41/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.76	
3.41/1.76	(0) CpxTRS
3.41/1.76	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.41/1.76	(2) TRS for Loop Detection
3.41/1.76	(3) DecreasingLoopProof [LOWER BOUND(ID), 69 ms]
3.41/1.76	(4) BEST
3.41/1.76	    (5) proven lower bound
3.41/1.76	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.41/1.76	        (7) BOUNDS(n^1, INF)
3.41/1.76	    (8) TRS for Loop Detection
3.41/1.76	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.41/1.76	        (10) BOUNDS(EXP, INF)
3.41/1.76	
3.41/1.76	
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(0)
3.41/1.76	Obligation:
3.41/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.76	
3.41/1.76	
3.41/1.76	The TRS R consists of the following rules:
3.41/1.76	
3.41/1.76	   dbl(0) -> 0
3.41/1.76	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.41/1.76	   dbls(nil) -> nil
3.41/1.76	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.41/1.76	   sel(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.41/1.76	   indx(nil, X) -> nil
3.41/1.76	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.41/1.76	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.41/1.76	   dbl1(0) -> 01
3.41/1.76	   dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
3.41/1.76	   sel1(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
3.41/1.76	   quote(0) -> 01
3.41/1.76	   quote(s(X)) -> s1(quote(activate(X)))
3.41/1.76	   quote(dbl(X)) -> dbl1(X)
3.41/1.76	   quote(sel(X, Y)) -> sel1(X, Y)
3.41/1.76	   s(X) -> n__s(X)
3.41/1.76	   dbl(X) -> n__dbl(X)
3.41/1.76	   dbls(X) -> n__dbls(X)
3.41/1.76	   sel(X1, X2) -> n__sel(X1, X2)
3.41/1.76	   indx(X1, X2) -> n__indx(X1, X2)
3.41/1.76	   from(X) -> n__from(X)
3.41/1.76	   activate(n__s(X)) -> s(X)
3.41/1.76	   activate(n__dbl(X)) -> dbl(X)
3.41/1.76	   activate(n__dbls(X)) -> dbls(X)
3.41/1.76	   activate(n__sel(X1, X2)) -> sel(X1, X2)
3.41/1.76	   activate(n__indx(X1, X2)) -> indx(X1, X2)
3.41/1.76	   activate(n__from(X)) -> from(X)
3.41/1.76	   activate(X) -> X
3.41/1.76	
3.41/1.76	S is empty.
3.41/1.76	Rewrite Strategy: FULL
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.41/1.76	Transformed a relative TRS into a decreasing-loop problem.
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(2)
3.41/1.76	Obligation:
3.41/1.76	Analyzing the following TRS for decreasing loops:
3.41/1.76	
3.41/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.76	
3.41/1.76	
3.41/1.76	The TRS R consists of the following rules:
3.41/1.76	
3.41/1.76	   dbl(0) -> 0
3.41/1.76	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.41/1.76	   dbls(nil) -> nil
3.41/1.76	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.41/1.76	   sel(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.41/1.76	   indx(nil, X) -> nil
3.41/1.76	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.41/1.76	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.41/1.76	   dbl1(0) -> 01
3.41/1.76	   dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
3.41/1.76	   sel1(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
3.41/1.76	   quote(0) -> 01
3.41/1.76	   quote(s(X)) -> s1(quote(activate(X)))
3.41/1.76	   quote(dbl(X)) -> dbl1(X)
3.41/1.76	   quote(sel(X, Y)) -> sel1(X, Y)
3.41/1.76	   s(X) -> n__s(X)
3.41/1.76	   dbl(X) -> n__dbl(X)
3.41/1.76	   dbls(X) -> n__dbls(X)
3.41/1.76	   sel(X1, X2) -> n__sel(X1, X2)
3.41/1.76	   indx(X1, X2) -> n__indx(X1, X2)
3.41/1.76	   from(X) -> n__from(X)
3.41/1.76	   activate(n__s(X)) -> s(X)
3.41/1.76	   activate(n__dbl(X)) -> dbl(X)
3.41/1.76	   activate(n__dbls(X)) -> dbls(X)
3.41/1.76	   activate(n__sel(X1, X2)) -> sel(X1, X2)
3.41/1.76	   activate(n__indx(X1, X2)) -> indx(X1, X2)
3.41/1.76	   activate(n__from(X)) -> from(X)
3.41/1.76	   activate(X) -> X
3.41/1.76	
3.41/1.76	S is empty.
3.41/1.76	Rewrite Strategy: FULL
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.41/1.76	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.41/1.76	
3.41/1.76	The rewrite sequence
3.41/1.76	
3.41/1.76	activate(n__dbls(cons(X1_0, Y2_0))) ->^+ cons(n__dbl(activate(X1_0)), n__dbls(activate(Y2_0)))
3.41/1.76	
3.41/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.41/1.76	
3.41/1.76	The pumping substitution is [X1_0 / n__dbls(cons(X1_0, Y2_0))].
3.41/1.76	
3.41/1.76	The result substitution is [ ].
3.41/1.76	
3.41/1.76	
3.41/1.76	
3.41/1.76	
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(4)
3.41/1.76	Complex Obligation (BEST)
3.41/1.76	
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(5)
3.41/1.76	Obligation:
3.41/1.76	Proved the lower bound n^1 for the following obligation:
3.41/1.76	
3.41/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.76	
3.41/1.76	
3.41/1.76	The TRS R consists of the following rules:
3.41/1.76	
3.41/1.76	   dbl(0) -> 0
3.41/1.76	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.41/1.76	   dbls(nil) -> nil
3.41/1.76	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.41/1.76	   sel(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.41/1.76	   indx(nil, X) -> nil
3.41/1.76	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.41/1.76	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.41/1.76	   dbl1(0) -> 01
3.41/1.76	   dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
3.41/1.76	   sel1(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
3.41/1.76	   quote(0) -> 01
3.41/1.76	   quote(s(X)) -> s1(quote(activate(X)))
3.41/1.76	   quote(dbl(X)) -> dbl1(X)
3.41/1.76	   quote(sel(X, Y)) -> sel1(X, Y)
3.41/1.76	   s(X) -> n__s(X)
3.41/1.76	   dbl(X) -> n__dbl(X)
3.41/1.76	   dbls(X) -> n__dbls(X)
3.41/1.76	   sel(X1, X2) -> n__sel(X1, X2)
3.41/1.76	   indx(X1, X2) -> n__indx(X1, X2)
3.41/1.76	   from(X) -> n__from(X)
3.41/1.76	   activate(n__s(X)) -> s(X)
3.41/1.76	   activate(n__dbl(X)) -> dbl(X)
3.41/1.76	   activate(n__dbls(X)) -> dbls(X)
3.41/1.76	   activate(n__sel(X1, X2)) -> sel(X1, X2)
3.41/1.76	   activate(n__indx(X1, X2)) -> indx(X1, X2)
3.41/1.76	   activate(n__from(X)) -> from(X)
3.41/1.76	   activate(X) -> X
3.41/1.76	
3.41/1.76	S is empty.
3.41/1.76	Rewrite Strategy: FULL
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(6) LowerBoundPropagationProof (FINISHED)
3.41/1.76	Propagated lower bound.
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(7)
3.41/1.76	BOUNDS(n^1, INF)
3.41/1.76	
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(8)
3.41/1.76	Obligation:
3.41/1.76	Analyzing the following TRS for decreasing loops:
3.41/1.76	
3.41/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.76	
3.41/1.76	
3.41/1.76	The TRS R consists of the following rules:
3.41/1.76	
3.41/1.76	   dbl(0) -> 0
3.41/1.76	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.41/1.76	   dbls(nil) -> nil
3.41/1.76	   dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y)))
3.41/1.76	   sel(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
3.41/1.76	   indx(nil, X) -> nil
3.41/1.76	   indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
3.41/1.76	   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
3.41/1.76	   dbl1(0) -> 01
3.41/1.76	   dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
3.41/1.76	   sel1(0, cons(X, Y)) -> activate(X)
3.41/1.76	   sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
3.41/1.76	   quote(0) -> 01
3.41/1.76	   quote(s(X)) -> s1(quote(activate(X)))
3.41/1.76	   quote(dbl(X)) -> dbl1(X)
3.41/1.76	   quote(sel(X, Y)) -> sel1(X, Y)
3.41/1.76	   s(X) -> n__s(X)
3.41/1.76	   dbl(X) -> n__dbl(X)
3.41/1.76	   dbls(X) -> n__dbls(X)
3.41/1.76	   sel(X1, X2) -> n__sel(X1, X2)
3.41/1.76	   indx(X1, X2) -> n__indx(X1, X2)
3.41/1.76	   from(X) -> n__from(X)
3.41/1.76	   activate(n__s(X)) -> s(X)
3.41/1.76	   activate(n__dbl(X)) -> dbl(X)
3.41/1.76	   activate(n__dbls(X)) -> dbls(X)
3.41/1.76	   activate(n__sel(X1, X2)) -> sel(X1, X2)
3.41/1.76	   activate(n__indx(X1, X2)) -> indx(X1, X2)
3.41/1.76	   activate(n__from(X)) -> from(X)
3.41/1.76	   activate(X) -> X
3.41/1.76	
3.41/1.76	S is empty.
3.41/1.76	Rewrite Strategy: FULL
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(9) DecreasingLoopProof (FINISHED)
3.41/1.76	The following loop(s) give(s) rise to the lower bound EXP:
3.41/1.76	
3.41/1.76	The rewrite sequence
3.41/1.76	
3.41/1.76	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.41/1.76	
3.41/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.41/1.76	
3.41/1.76	The pumping substitution is [X / n__from(X)].
3.41/1.76	
3.41/1.76	The result substitution is [ ].
3.41/1.76	
3.41/1.76	
3.41/1.76	
3.41/1.76	The rewrite sequence
3.41/1.76	
3.41/1.76	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.41/1.76	
3.41/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.41/1.76	
3.41/1.76	The pumping substitution is [X / n__from(X)].
3.41/1.76	
3.41/1.76	The result substitution is [ ].
3.41/1.76	
3.41/1.76	
3.41/1.76	
3.41/1.76	
3.41/1.76	----------------------------------------
3.41/1.76	
3.41/1.76	(10)
3.41/1.76	BOUNDS(EXP, INF)
3.66/1.80	EOF