1130.02/291.56	WORST_CASE(Omega(n^1), ?)
1137.31/293.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1137.31/293.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.31/293.43	
1137.31/293.43	(0) CpxTRS
1137.31/293.43	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1137.31/293.43	(2) TRS for Loop Detection
1137.31/293.43	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1137.31/293.43	(4) BEST
1137.31/293.43	    (5) proven lower bound
1137.31/293.43	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1137.31/293.43	        (7) BOUNDS(n^1, INF)
1137.31/293.43	    (8) TRS for Loop Detection
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(0)
1137.31/293.43	Obligation:
1137.31/293.43	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	The TRS R consists of the following rules:
1137.31/293.43	
1137.31/293.43	   a__dbl(0) -> 0
1137.31/293.43	   a__dbl(s(X)) -> s(s(dbl(X)))
1137.31/293.43	   a__dbls(nil) -> nil
1137.31/293.43	   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
1137.31/293.43	   a__sel(0, cons(X, Y)) -> mark(X)
1137.31/293.43	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1137.31/293.43	   a__indx(nil, X) -> nil
1137.31/293.43	   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
1137.31/293.43	   a__from(X) -> cons(X, from(s(X)))
1137.31/293.43	   mark(dbl(X)) -> a__dbl(mark(X))
1137.31/293.43	   mark(dbls(X)) -> a__dbls(mark(X))
1137.31/293.43	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1137.31/293.43	   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
1137.31/293.43	   mark(from(X)) -> a__from(X)
1137.31/293.43	   mark(0) -> 0
1137.31/293.43	   mark(s(X)) -> s(X)
1137.31/293.43	   mark(nil) -> nil
1137.31/293.43	   mark(cons(X1, X2)) -> cons(X1, X2)
1137.31/293.43	   a__dbl(X) -> dbl(X)
1137.31/293.43	   a__dbls(X) -> dbls(X)
1137.31/293.43	   a__sel(X1, X2) -> sel(X1, X2)
1137.31/293.43	   a__indx(X1, X2) -> indx(X1, X2)
1137.31/293.43	   a__from(X) -> from(X)
1137.31/293.43	
1137.31/293.43	S is empty.
1137.31/293.43	Rewrite Strategy: INNERMOST
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1137.31/293.43	Transformed a relative TRS into a decreasing-loop problem.
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(2)
1137.31/293.43	Obligation:
1137.31/293.43	Analyzing the following TRS for decreasing loops:
1137.31/293.43	
1137.31/293.43	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	The TRS R consists of the following rules:
1137.31/293.43	
1137.31/293.43	   a__dbl(0) -> 0
1137.31/293.43	   a__dbl(s(X)) -> s(s(dbl(X)))
1137.31/293.43	   a__dbls(nil) -> nil
1137.31/293.43	   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
1137.31/293.43	   a__sel(0, cons(X, Y)) -> mark(X)
1137.31/293.43	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1137.31/293.43	   a__indx(nil, X) -> nil
1137.31/293.43	   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
1137.31/293.43	   a__from(X) -> cons(X, from(s(X)))
1137.31/293.43	   mark(dbl(X)) -> a__dbl(mark(X))
1137.31/293.43	   mark(dbls(X)) -> a__dbls(mark(X))
1137.31/293.43	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1137.31/293.43	   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
1137.31/293.43	   mark(from(X)) -> a__from(X)
1137.31/293.43	   mark(0) -> 0
1137.31/293.43	   mark(s(X)) -> s(X)
1137.31/293.43	   mark(nil) -> nil
1137.31/293.43	   mark(cons(X1, X2)) -> cons(X1, X2)
1137.31/293.43	   a__dbl(X) -> dbl(X)
1137.31/293.43	   a__dbls(X) -> dbls(X)
1137.31/293.43	   a__sel(X1, X2) -> sel(X1, X2)
1137.31/293.43	   a__indx(X1, X2) -> indx(X1, X2)
1137.31/293.43	   a__from(X) -> from(X)
1137.31/293.43	
1137.31/293.43	S is empty.
1137.31/293.43	Rewrite Strategy: INNERMOST
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(3) DecreasingLoopProof (LOWER BOUND(ID))
1137.31/293.43	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1137.31/293.43	
1137.31/293.43	The rewrite sequence
1137.31/293.43	
1137.31/293.43	mark(indx(X1, X2)) ->^+ a__indx(mark(X1), X2)
1137.31/293.43	
1137.31/293.43	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1137.31/293.43	
1137.31/293.43	The pumping substitution is [X1 / indx(X1, X2)].
1137.31/293.43	
1137.31/293.43	The result substitution is [ ].
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(4)
1137.31/293.43	Complex Obligation (BEST)
1137.31/293.43	
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(5)
1137.31/293.43	Obligation:
1137.31/293.43	Proved the lower bound n^1 for the following obligation:
1137.31/293.43	
1137.31/293.43	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	The TRS R consists of the following rules:
1137.31/293.43	
1137.31/293.43	   a__dbl(0) -> 0
1137.31/293.43	   a__dbl(s(X)) -> s(s(dbl(X)))
1137.31/293.43	   a__dbls(nil) -> nil
1137.31/293.43	   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
1137.31/293.43	   a__sel(0, cons(X, Y)) -> mark(X)
1137.31/293.43	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1137.31/293.43	   a__indx(nil, X) -> nil
1137.31/293.43	   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
1137.31/293.43	   a__from(X) -> cons(X, from(s(X)))
1137.31/293.43	   mark(dbl(X)) -> a__dbl(mark(X))
1137.31/293.43	   mark(dbls(X)) -> a__dbls(mark(X))
1137.31/293.43	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1137.31/293.43	   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
1137.31/293.43	   mark(from(X)) -> a__from(X)
1137.31/293.43	   mark(0) -> 0
1137.31/293.43	   mark(s(X)) -> s(X)
1137.31/293.43	   mark(nil) -> nil
1137.31/293.43	   mark(cons(X1, X2)) -> cons(X1, X2)
1137.31/293.43	   a__dbl(X) -> dbl(X)
1137.31/293.43	   a__dbls(X) -> dbls(X)
1137.31/293.43	   a__sel(X1, X2) -> sel(X1, X2)
1137.31/293.43	   a__indx(X1, X2) -> indx(X1, X2)
1137.31/293.43	   a__from(X) -> from(X)
1137.31/293.43	
1137.31/293.43	S is empty.
1137.31/293.43	Rewrite Strategy: INNERMOST
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(6) LowerBoundPropagationProof (FINISHED)
1137.31/293.43	Propagated lower bound.
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(7)
1137.31/293.43	BOUNDS(n^1, INF)
1137.31/293.43	
1137.31/293.43	----------------------------------------
1137.31/293.43	
1137.31/293.43	(8)
1137.31/293.43	Obligation:
1137.31/293.43	Analyzing the following TRS for decreasing loops:
1137.31/293.43	
1137.31/293.43	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.31/293.43	
1137.31/293.43	
1137.31/293.43	The TRS R consists of the following rules:
1137.31/293.43	
1137.31/293.43	   a__dbl(0) -> 0
1137.31/293.43	   a__dbl(s(X)) -> s(s(dbl(X)))
1137.31/293.43	   a__dbls(nil) -> nil
1137.31/293.43	   a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
1137.31/293.43	   a__sel(0, cons(X, Y)) -> mark(X)
1137.31/293.43	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1137.31/293.43	   a__indx(nil, X) -> nil
1137.31/293.43	   a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
1137.31/293.43	   a__from(X) -> cons(X, from(s(X)))
1137.31/293.43	   mark(dbl(X)) -> a__dbl(mark(X))
1137.31/293.43	   mark(dbls(X)) -> a__dbls(mark(X))
1137.31/293.43	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1137.31/293.43	   mark(indx(X1, X2)) -> a__indx(mark(X1), X2)
1137.31/293.43	   mark(from(X)) -> a__from(X)
1137.31/293.43	   mark(0) -> 0
1137.31/293.43	   mark(s(X)) -> s(X)
1137.31/293.43	   mark(nil) -> nil
1137.31/293.43	   mark(cons(X1, X2)) -> cons(X1, X2)
1137.31/293.43	   a__dbl(X) -> dbl(X)
1137.31/293.43	   a__dbls(X) -> dbls(X)
1137.31/293.43	   a__sel(X1, X2) -> sel(X1, X2)
1137.31/293.43	   a__indx(X1, X2) -> indx(X1, X2)
1137.31/293.43	   a__from(X) -> from(X)
1137.31/293.43	
1137.31/293.43	S is empty.
1137.31/293.43	Rewrite Strategy: INNERMOST
1137.49/293.50	EOF