3.25/1.67	WORST_CASE(NON_POLY, ?)
3.25/1.68	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.25/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.25/1.68	
3.25/1.68	
3.25/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.25/1.68	
3.25/1.68	(0) CpxTRS
3.25/1.68	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.25/1.68	(2) TRS for Loop Detection
3.25/1.68	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.25/1.68	(4) BEST
3.25/1.68	    (5) proven lower bound
3.25/1.68	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.25/1.68	        (7) BOUNDS(n^1, INF)
3.25/1.68	    (8) TRS for Loop Detection
3.25/1.68	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.25/1.68	        (10) BOUNDS(INF, INF)
3.25/1.68	
3.25/1.68	
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(0)
3.25/1.68	Obligation:
3.25/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.25/1.68	
3.25/1.68	
3.25/1.68	The TRS R consists of the following rules:
3.25/1.68	
3.25/1.68	   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
3.25/1.68	   badd(x, Nil) -> x
3.25/1.68	   goal(x, y) -> badd(x, y)
3.25/1.68	
3.25/1.68	S is empty.
3.25/1.68	Rewrite Strategy: INNERMOST
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.25/1.68	Transformed a relative TRS into a decreasing-loop problem.
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(2)
3.25/1.68	Obligation:
3.25/1.68	Analyzing the following TRS for decreasing loops:
3.25/1.68	
3.25/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.25/1.68	
3.25/1.68	
3.25/1.68	The TRS R consists of the following rules:
3.25/1.68	
3.25/1.68	   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
3.25/1.68	   badd(x, Nil) -> x
3.25/1.68	   goal(x, y) -> badd(x, y)
3.25/1.68	
3.25/1.68	S is empty.
3.25/1.68	Rewrite Strategy: INNERMOST
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.25/1.68	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.25/1.68	
3.25/1.68	The rewrite sequence
3.25/1.68	
3.25/1.68	badd(x', Cons(x, xs)) ->^+ badd(Cons(Nil, Nil), badd(x', xs))
3.25/1.68	
3.25/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.25/1.68	
3.25/1.68	The pumping substitution is [xs / Cons(x, xs)].
3.25/1.68	
3.25/1.68	The result substitution is [ ].
3.25/1.68	
3.25/1.68	
3.25/1.68	
3.25/1.68	
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(4)
3.25/1.68	Complex Obligation (BEST)
3.25/1.68	
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(5)
3.25/1.68	Obligation:
3.25/1.68	Proved the lower bound n^1 for the following obligation:
3.25/1.68	
3.25/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.25/1.68	
3.25/1.68	
3.25/1.68	The TRS R consists of the following rules:
3.25/1.68	
3.25/1.68	   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
3.25/1.68	   badd(x, Nil) -> x
3.25/1.68	   goal(x, y) -> badd(x, y)
3.25/1.68	
3.25/1.68	S is empty.
3.25/1.68	Rewrite Strategy: INNERMOST
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(6) LowerBoundPropagationProof (FINISHED)
3.25/1.68	Propagated lower bound.
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(7)
3.25/1.68	BOUNDS(n^1, INF)
3.25/1.68	
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(8)
3.25/1.68	Obligation:
3.25/1.68	Analyzing the following TRS for decreasing loops:
3.25/1.68	
3.25/1.68	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.25/1.68	
3.25/1.68	
3.25/1.68	The TRS R consists of the following rules:
3.25/1.68	
3.25/1.68	   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
3.25/1.68	   badd(x, Nil) -> x
3.25/1.68	   goal(x, y) -> badd(x, y)
3.25/1.68	
3.25/1.68	S is empty.
3.25/1.68	Rewrite Strategy: INNERMOST
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(9) InfiniteLowerBoundProof (FINISHED)
3.25/1.68	The following loop proves infinite runtime complexity:
3.25/1.68	
3.25/1.68	The rewrite sequence
3.25/1.68	
3.25/1.68	badd(Cons(x2_0, Nil), Cons(x, Nil)) ->^+ badd(Cons(Nil, Nil), Cons(Nil, Nil))
3.25/1.68	
3.25/1.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.25/1.68	
3.25/1.68	The pumping substitution is [ ].
3.25/1.68	
3.25/1.68	The result substitution is [x2_0 / Nil, x / Nil].
3.25/1.68	
3.25/1.68	
3.25/1.68	
3.25/1.68	
3.25/1.68	----------------------------------------
3.25/1.68	
3.25/1.68	(10)
3.25/1.68	BOUNDS(INF, INF)
3.49/1.70	EOF