1063.43/291.59	WORST_CASE(Omega(n^1), ?)
1063.43/291.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1063.43/291.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1063.43/291.60	
1063.43/291.60	(0) CpxTRS
1063.43/291.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1063.43/291.60	(2) TRS for Loop Detection
1063.43/291.60	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1063.43/291.60	(4) BEST
1063.43/291.60	    (5) proven lower bound
1063.43/291.60	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1063.43/291.60	        (7) BOUNDS(n^1, INF)
1063.43/291.60	    (8) TRS for Loop Detection
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(0)
1063.43/291.60	Obligation:
1063.43/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	The TRS R consists of the following rules:
1063.43/291.60	
1063.43/291.60	   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
1063.43/291.60	   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
1063.43/291.60	   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
1063.43/291.60	   goal(m, n) -> ack(m, n)
1063.43/291.60	
1063.43/291.60	S is empty.
1063.43/291.60	Rewrite Strategy: INNERMOST
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1063.43/291.60	Transformed a relative TRS into a decreasing-loop problem.
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(2)
1063.43/291.60	Obligation:
1063.43/291.60	Analyzing the following TRS for decreasing loops:
1063.43/291.60	
1063.43/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	The TRS R consists of the following rules:
1063.43/291.60	
1063.43/291.60	   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
1063.43/291.60	   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
1063.43/291.60	   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
1063.43/291.60	   goal(m, n) -> ack(m, n)
1063.43/291.60	
1063.43/291.60	S is empty.
1063.43/291.60	Rewrite Strategy: INNERMOST
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(3) DecreasingLoopProof (LOWER BOUND(ID))
1063.43/291.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1063.43/291.60	
1063.43/291.60	The rewrite sequence
1063.43/291.60	
1063.43/291.60	ack(Cons(x', xs'), Cons(x, xs)) ->^+ ack(xs', ack(Cons(x', xs'), xs))
1063.43/291.60	
1063.43/291.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
1063.43/291.60	
1063.43/291.60	The pumping substitution is [xs / Cons(x, xs)].
1063.43/291.60	
1063.43/291.60	The result substitution is [ ].
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(4)
1063.43/291.60	Complex Obligation (BEST)
1063.43/291.60	
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(5)
1063.43/291.60	Obligation:
1063.43/291.60	Proved the lower bound n^1 for the following obligation:
1063.43/291.60	
1063.43/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	The TRS R consists of the following rules:
1063.43/291.60	
1063.43/291.60	   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
1063.43/291.60	   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
1063.43/291.60	   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
1063.43/291.60	   goal(m, n) -> ack(m, n)
1063.43/291.60	
1063.43/291.60	S is empty.
1063.43/291.60	Rewrite Strategy: INNERMOST
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(6) LowerBoundPropagationProof (FINISHED)
1063.43/291.60	Propagated lower bound.
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(7)
1063.43/291.60	BOUNDS(n^1, INF)
1063.43/291.60	
1063.43/291.60	----------------------------------------
1063.43/291.60	
1063.43/291.60	(8)
1063.43/291.60	Obligation:
1063.43/291.60	Analyzing the following TRS for decreasing loops:
1063.43/291.60	
1063.43/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1063.43/291.60	
1063.43/291.60	
1063.43/291.60	The TRS R consists of the following rules:
1063.43/291.60	
1063.43/291.60	   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
1063.43/291.60	   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
1063.43/291.60	   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
1063.43/291.60	   goal(m, n) -> ack(m, n)
1063.43/291.60	
1063.43/291.60	S is empty.
1063.43/291.60	Rewrite Strategy: INNERMOST
1063.59/291.64	EOF