310.67/291.53	WORST_CASE(Omega(n^1), ?)
310.73/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
310.73/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
310.73/291.54	
310.73/291.54	
310.73/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.73/291.54	
310.73/291.54	(0) CpxTRS
310.73/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
310.73/291.54	(2) TRS for Loop Detection
310.73/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
310.73/291.54	(4) BEST
310.73/291.54	    (5) proven lower bound
310.73/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
310.73/291.54	        (7) BOUNDS(n^1, INF)
310.73/291.54	    (8) TRS for Loop Detection
310.73/291.54	
310.73/291.54	
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(0)
310.73/291.54	Obligation:
310.73/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.73/291.54	
310.73/291.54	
310.73/291.54	The TRS R consists of the following rules:
310.73/291.54	
310.73/291.54	   eq(0, 0) -> true
310.73/291.54	   eq(0, s(m)) -> false
310.73/291.54	   eq(s(n), 0) -> false
310.73/291.54	   eq(s(n), s(m)) -> eq(n, m)
310.73/291.54	   le(0, m) -> true
310.73/291.54	   le(s(n), 0) -> false
310.73/291.54	   le(s(n), s(m)) -> le(n, m)
310.73/291.54	   min(cons(x, nil)) -> x
310.73/291.54	   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
310.73/291.54	   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
310.73/291.54	   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
310.73/291.54	   replace(n, m, nil) -> nil
310.73/291.54	   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
310.73/291.54	   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
310.73/291.54	   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
310.73/291.54	   empty(nil) -> true
310.73/291.54	   empty(cons(n, x)) -> false
310.73/291.54	   head(cons(n, x)) -> n
310.73/291.54	   tail(nil) -> nil
310.73/291.54	   tail(cons(n, x)) -> x
310.73/291.54	   sort(x) -> sortIter(x, nil)
310.73/291.54	   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
310.73/291.54	   if(true, x, y, z) -> y
310.73/291.54	   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)
310.73/291.54	
310.73/291.54	S is empty.
310.73/291.54	Rewrite Strategy: FULL
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
310.73/291.54	Transformed a relative TRS into a decreasing-loop problem.
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(2)
310.73/291.54	Obligation:
310.73/291.54	Analyzing the following TRS for decreasing loops:
310.73/291.54	
310.73/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.73/291.54	
310.73/291.54	
310.73/291.54	The TRS R consists of the following rules:
310.73/291.54	
310.73/291.54	   eq(0, 0) -> true
310.73/291.54	   eq(0, s(m)) -> false
310.73/291.54	   eq(s(n), 0) -> false
310.73/291.54	   eq(s(n), s(m)) -> eq(n, m)
310.73/291.54	   le(0, m) -> true
310.73/291.54	   le(s(n), 0) -> false
310.73/291.54	   le(s(n), s(m)) -> le(n, m)
310.73/291.54	   min(cons(x, nil)) -> x
310.73/291.54	   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
310.73/291.54	   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
310.73/291.54	   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
310.73/291.54	   replace(n, m, nil) -> nil
310.73/291.54	   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
310.73/291.54	   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
310.73/291.54	   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
310.73/291.54	   empty(nil) -> true
310.73/291.54	   empty(cons(n, x)) -> false
310.73/291.54	   head(cons(n, x)) -> n
310.73/291.54	   tail(nil) -> nil
310.73/291.54	   tail(cons(n, x)) -> x
310.73/291.54	   sort(x) -> sortIter(x, nil)
310.73/291.54	   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
310.73/291.54	   if(true, x, y, z) -> y
310.73/291.54	   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)
310.73/291.54	
310.73/291.54	S is empty.
310.73/291.54	Rewrite Strategy: FULL
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
310.73/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
310.73/291.54	
310.73/291.54	The rewrite sequence
310.73/291.54	
310.73/291.54	eq(s(n), s(m)) ->^+ eq(n, m)
310.73/291.54	
310.73/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
310.73/291.54	
310.73/291.54	The pumping substitution is [n / s(n), m / s(m)].
310.73/291.54	
310.73/291.54	The result substitution is [ ].
310.73/291.54	
310.73/291.54	
310.73/291.54	
310.73/291.54	
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(4)
310.73/291.54	Complex Obligation (BEST)
310.73/291.54	
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(5)
310.73/291.54	Obligation:
310.73/291.54	Proved the lower bound n^1 for the following obligation:
310.73/291.54	
310.73/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.73/291.54	
310.73/291.54	
310.73/291.54	The TRS R consists of the following rules:
310.73/291.54	
310.73/291.54	   eq(0, 0) -> true
310.73/291.54	   eq(0, s(m)) -> false
310.73/291.54	   eq(s(n), 0) -> false
310.73/291.54	   eq(s(n), s(m)) -> eq(n, m)
310.73/291.54	   le(0, m) -> true
310.73/291.54	   le(s(n), 0) -> false
310.73/291.54	   le(s(n), s(m)) -> le(n, m)
310.73/291.54	   min(cons(x, nil)) -> x
310.73/291.54	   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
310.73/291.54	   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
310.73/291.54	   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
310.73/291.54	   replace(n, m, nil) -> nil
310.73/291.54	   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
310.73/291.54	   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
310.73/291.54	   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
310.73/291.54	   empty(nil) -> true
310.73/291.54	   empty(cons(n, x)) -> false
310.73/291.54	   head(cons(n, x)) -> n
310.73/291.54	   tail(nil) -> nil
310.73/291.54	   tail(cons(n, x)) -> x
310.73/291.54	   sort(x) -> sortIter(x, nil)
310.73/291.54	   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
310.73/291.54	   if(true, x, y, z) -> y
310.73/291.54	   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)
310.73/291.54	
310.73/291.54	S is empty.
310.73/291.54	Rewrite Strategy: FULL
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(6) LowerBoundPropagationProof (FINISHED)
310.73/291.54	Propagated lower bound.
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(7)
310.73/291.54	BOUNDS(n^1, INF)
310.73/291.54	
310.73/291.54	----------------------------------------
310.73/291.54	
310.73/291.54	(8)
310.73/291.54	Obligation:
310.73/291.54	Analyzing the following TRS for decreasing loops:
310.73/291.54	
310.73/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.73/291.54	
310.73/291.54	
310.73/291.54	The TRS R consists of the following rules:
310.73/291.54	
310.73/291.54	   eq(0, 0) -> true
310.73/291.54	   eq(0, s(m)) -> false
310.73/291.54	   eq(s(n), 0) -> false
310.73/291.54	   eq(s(n), s(m)) -> eq(n, m)
310.73/291.54	   le(0, m) -> true
310.73/291.54	   le(s(n), 0) -> false
310.73/291.54	   le(s(n), s(m)) -> le(n, m)
310.73/291.54	   min(cons(x, nil)) -> x
310.73/291.54	   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
310.73/291.54	   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
310.73/291.54	   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
310.73/291.54	   replace(n, m, nil) -> nil
310.73/291.54	   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
310.73/291.54	   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
310.73/291.54	   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
310.73/291.54	   empty(nil) -> true
310.73/291.54	   empty(cons(n, x)) -> false
310.73/291.54	   head(cons(n, x)) -> n
310.73/291.54	   tail(nil) -> nil
310.73/291.54	   tail(cons(n, x)) -> x
310.73/291.54	   sort(x) -> sortIter(x, nil)
310.73/291.54	   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
310.73/291.54	   if(true, x, y, z) -> y
310.73/291.54	   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)
310.73/291.54	
310.73/291.54	S is empty.
310.73/291.54	Rewrite Strategy: FULL
310.77/291.56	EOF