314.97/291.54	WORST_CASE(Omega(n^1), ?)
314.97/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
314.97/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
314.97/291.54	
314.97/291.54	
314.97/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.97/291.54	
314.97/291.54	(0) CpxTRS
314.97/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
314.97/291.54	(2) TRS for Loop Detection
314.97/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
314.97/291.54	(4) BEST
314.97/291.54	    (5) proven lower bound
314.97/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
314.97/291.54	        (7) BOUNDS(n^1, INF)
314.97/291.54	    (8) TRS for Loop Detection
314.97/291.54	
314.97/291.54	
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(0)
314.97/291.54	Obligation:
314.97/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.97/291.54	
314.97/291.54	
314.97/291.54	The TRS R consists of the following rules:
314.97/291.54	
314.97/291.54	   sort(nil) -> nil
314.97/291.54	   sort(cons(x, y)) -> insert(x, sort(y))
314.97/291.54	   insert(x, nil) -> cons(x, nil)
314.97/291.54	   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
314.97/291.54	   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
314.97/291.54	   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
314.97/291.54	   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)
314.97/291.54	
314.97/291.54	S is empty.
314.97/291.54	Rewrite Strategy: FULL
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
314.97/291.54	Transformed a relative TRS into a decreasing-loop problem.
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(2)
314.97/291.54	Obligation:
314.97/291.54	Analyzing the following TRS for decreasing loops:
314.97/291.54	
314.97/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.97/291.54	
314.97/291.54	
314.97/291.54	The TRS R consists of the following rules:
314.97/291.54	
314.97/291.54	   sort(nil) -> nil
314.97/291.54	   sort(cons(x, y)) -> insert(x, sort(y))
314.97/291.54	   insert(x, nil) -> cons(x, nil)
314.97/291.54	   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
314.97/291.54	   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
314.97/291.54	   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
314.97/291.54	   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)
314.97/291.54	
314.97/291.54	S is empty.
314.97/291.54	Rewrite Strategy: FULL
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
314.97/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
314.97/291.54	
314.97/291.54	The rewrite sequence
314.97/291.54	
314.97/291.54	sort(cons(x, y)) ->^+ insert(x, sort(y))
314.97/291.54	
314.97/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
314.97/291.54	
314.97/291.54	The pumping substitution is [y / cons(x, y)].
314.97/291.54	
314.97/291.54	The result substitution is [ ].
314.97/291.54	
314.97/291.54	
314.97/291.54	
314.97/291.54	
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(4)
314.97/291.54	Complex Obligation (BEST)
314.97/291.54	
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(5)
314.97/291.54	Obligation:
314.97/291.54	Proved the lower bound n^1 for the following obligation:
314.97/291.54	
314.97/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.97/291.54	
314.97/291.54	
314.97/291.54	The TRS R consists of the following rules:
314.97/291.54	
314.97/291.54	   sort(nil) -> nil
314.97/291.54	   sort(cons(x, y)) -> insert(x, sort(y))
314.97/291.54	   insert(x, nil) -> cons(x, nil)
314.97/291.54	   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
314.97/291.54	   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
314.97/291.54	   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
314.97/291.54	   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)
314.97/291.54	
314.97/291.54	S is empty.
314.97/291.54	Rewrite Strategy: FULL
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(6) LowerBoundPropagationProof (FINISHED)
314.97/291.54	Propagated lower bound.
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(7)
314.97/291.54	BOUNDS(n^1, INF)
314.97/291.54	
314.97/291.54	----------------------------------------
314.97/291.54	
314.97/291.54	(8)
314.97/291.54	Obligation:
314.97/291.54	Analyzing the following TRS for decreasing loops:
314.97/291.54	
314.97/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.97/291.54	
314.97/291.54	
314.97/291.54	The TRS R consists of the following rules:
314.97/291.54	
314.97/291.54	   sort(nil) -> nil
314.97/291.54	   sort(cons(x, y)) -> insert(x, sort(y))
314.97/291.54	   insert(x, nil) -> cons(x, nil)
314.97/291.54	   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
314.97/291.54	   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
314.97/291.54	   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
314.97/291.54	   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)
314.97/291.54	
314.97/291.54	S is empty.
314.97/291.54	Rewrite Strategy: FULL
314.97/291.57	EOF