1138.38/291.51	WORST_CASE(Omega(n^1), ?)
1138.80/291.62	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1138.80/291.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1138.80/291.62	
1138.80/291.62	(0) CpxTRS
1138.80/291.62	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1138.80/291.62	(2) TRS for Loop Detection
1138.80/291.62	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1138.80/291.62	(4) BEST
1138.80/291.62	    (5) proven lower bound
1138.80/291.62	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1138.80/291.62	        (7) BOUNDS(n^1, INF)
1138.80/291.62	    (8) TRS for Loop Detection
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(0)
1138.80/291.62	Obligation:
1138.80/291.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	The TRS R consists of the following rules:
1138.80/291.62	
1138.80/291.62	   isEmpty(empty) -> true
1138.80/291.62	   isEmpty(node(l, x, r)) -> false
1138.80/291.62	   left(empty) -> empty
1138.80/291.62	   left(node(l, x, r)) -> l
1138.80/291.62	   right(empty) -> empty
1138.80/291.62	   right(node(l, x, r)) -> r
1138.80/291.62	   elem(node(l, x, r)) -> x
1138.80/291.62	   append(nil, x) -> cons(x, nil)
1138.80/291.62	   append(cons(y, ys), x) -> cons(y, append(ys, x))
1138.80/291.62	   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
1138.80/291.62	   if(true, b, n, m, xs, ys) -> xs
1138.80/291.62	   if(false, false, n, m, xs, ys) -> listify(m, xs)
1138.80/291.62	   if(false, true, n, m, xs, ys) -> listify(n, ys)
1138.80/291.62	   toList(n) -> listify(n, nil)
1138.80/291.62	
1138.80/291.62	S is empty.
1138.80/291.62	Rewrite Strategy: INNERMOST
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1138.80/291.62	Transformed a relative TRS into a decreasing-loop problem.
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(2)
1138.80/291.62	Obligation:
1138.80/291.62	Analyzing the following TRS for decreasing loops:
1138.80/291.62	
1138.80/291.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	The TRS R consists of the following rules:
1138.80/291.62	
1138.80/291.62	   isEmpty(empty) -> true
1138.80/291.62	   isEmpty(node(l, x, r)) -> false
1138.80/291.62	   left(empty) -> empty
1138.80/291.62	   left(node(l, x, r)) -> l
1138.80/291.62	   right(empty) -> empty
1138.80/291.62	   right(node(l, x, r)) -> r
1138.80/291.62	   elem(node(l, x, r)) -> x
1138.80/291.62	   append(nil, x) -> cons(x, nil)
1138.80/291.62	   append(cons(y, ys), x) -> cons(y, append(ys, x))
1138.80/291.62	   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
1138.80/291.62	   if(true, b, n, m, xs, ys) -> xs
1138.80/291.62	   if(false, false, n, m, xs, ys) -> listify(m, xs)
1138.80/291.62	   if(false, true, n, m, xs, ys) -> listify(n, ys)
1138.80/291.62	   toList(n) -> listify(n, nil)
1138.80/291.62	
1138.80/291.62	S is empty.
1138.80/291.62	Rewrite Strategy: INNERMOST
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(3) DecreasingLoopProof (LOWER BOUND(ID))
1138.80/291.62	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1138.80/291.62	
1138.80/291.62	The rewrite sequence
1138.80/291.62	
1138.80/291.62	append(cons(y, ys), x) ->^+ cons(y, append(ys, x))
1138.80/291.62	
1138.80/291.62	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
1138.80/291.62	
1138.80/291.62	The pumping substitution is [ys / cons(y, ys)].
1138.80/291.62	
1138.80/291.62	The result substitution is [ ].
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(4)
1138.80/291.62	Complex Obligation (BEST)
1138.80/291.62	
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(5)
1138.80/291.62	Obligation:
1138.80/291.62	Proved the lower bound n^1 for the following obligation:
1138.80/291.62	
1138.80/291.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	The TRS R consists of the following rules:
1138.80/291.62	
1138.80/291.62	   isEmpty(empty) -> true
1138.80/291.62	   isEmpty(node(l, x, r)) -> false
1138.80/291.62	   left(empty) -> empty
1138.80/291.62	   left(node(l, x, r)) -> l
1138.80/291.62	   right(empty) -> empty
1138.80/291.62	   right(node(l, x, r)) -> r
1138.80/291.62	   elem(node(l, x, r)) -> x
1138.80/291.62	   append(nil, x) -> cons(x, nil)
1138.80/291.62	   append(cons(y, ys), x) -> cons(y, append(ys, x))
1138.80/291.62	   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
1138.80/291.62	   if(true, b, n, m, xs, ys) -> xs
1138.80/291.62	   if(false, false, n, m, xs, ys) -> listify(m, xs)
1138.80/291.62	   if(false, true, n, m, xs, ys) -> listify(n, ys)
1138.80/291.62	   toList(n) -> listify(n, nil)
1138.80/291.62	
1138.80/291.62	S is empty.
1138.80/291.62	Rewrite Strategy: INNERMOST
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(6) LowerBoundPropagationProof (FINISHED)
1138.80/291.62	Propagated lower bound.
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(7)
1138.80/291.62	BOUNDS(n^1, INF)
1138.80/291.62	
1138.80/291.62	----------------------------------------
1138.80/291.62	
1138.80/291.62	(8)
1138.80/291.62	Obligation:
1138.80/291.62	Analyzing the following TRS for decreasing loops:
1138.80/291.62	
1138.80/291.62	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1138.80/291.62	
1138.80/291.62	
1138.80/291.62	The TRS R consists of the following rules:
1138.80/291.62	
1138.80/291.62	   isEmpty(empty) -> true
1138.80/291.62	   isEmpty(node(l, x, r)) -> false
1138.80/291.62	   left(empty) -> empty
1138.80/291.62	   left(node(l, x, r)) -> l
1138.80/291.62	   right(empty) -> empty
1138.80/291.62	   right(node(l, x, r)) -> r
1138.80/291.62	   elem(node(l, x, r)) -> x
1138.80/291.62	   append(nil, x) -> cons(x, nil)
1138.80/291.62	   append(cons(y, ys), x) -> cons(y, append(ys, x))
1138.80/291.62	   listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n))
1138.80/291.62	   if(true, b, n, m, xs, ys) -> xs
1138.80/291.62	   if(false, false, n, m, xs, ys) -> listify(m, xs)
1138.80/291.62	   if(false, true, n, m, xs, ys) -> listify(n, ys)
1138.80/291.62	   toList(n) -> listify(n, nil)
1138.80/291.62	
1138.80/291.62	S is empty.
1138.80/291.62	Rewrite Strategy: INNERMOST
1139.07/291.69	EOF