3.55/2.22	WORST_CASE(NON_POLY, ?)
3.55/2.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.55/2.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.55/2.22	
3.55/2.22	
3.55/2.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.55/2.22	
3.55/2.22	(0) CpxTRS
3.55/2.22	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.55/2.22	(2) TRS for Loop Detection
3.55/2.22	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.55/2.22	(4) BEST
3.55/2.22	    (5) proven lower bound
3.55/2.22	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.55/2.22	        (7) BOUNDS(n^1, INF)
3.55/2.22	    (8) TRS for Loop Detection
3.55/2.22	        (9) InfiniteLowerBoundProof [FINISHED, 1 ms]
3.55/2.22	        (10) BOUNDS(INF, INF)
3.55/2.22	
3.55/2.22	
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(0)
3.55/2.22	Obligation:
3.55/2.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.55/2.22	
3.55/2.22	
3.55/2.22	The TRS R consists of the following rules:
3.55/2.22	
3.55/2.22	   f(s(s(s(s(s(s(s(s(x)))))))), y, y) -> f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
3.55/2.22	   id(s(x)) -> s(id(x))
3.55/2.22	   id(0) -> 0
3.55/2.22	
3.55/2.22	S is empty.
3.55/2.22	Rewrite Strategy: FULL
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.55/2.22	Transformed a relative TRS into a decreasing-loop problem.
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(2)
3.55/2.22	Obligation:
3.55/2.22	Analyzing the following TRS for decreasing loops:
3.55/2.22	
3.55/2.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.55/2.22	
3.55/2.22	
3.55/2.22	The TRS R consists of the following rules:
3.55/2.22	
3.55/2.22	   f(s(s(s(s(s(s(s(s(x)))))))), y, y) -> f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
3.55/2.22	   id(s(x)) -> s(id(x))
3.55/2.22	   id(0) -> 0
3.55/2.22	
3.55/2.22	S is empty.
3.55/2.22	Rewrite Strategy: FULL
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.55/2.22	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.55/2.22	
3.55/2.22	The rewrite sequence
3.55/2.22	
3.55/2.22	id(s(x)) ->^+ s(id(x))
3.55/2.22	
3.55/2.22	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.55/2.22	
3.55/2.22	The pumping substitution is [x / s(x)].
3.55/2.22	
3.55/2.22	The result substitution is [ ].
3.55/2.22	
3.55/2.22	
3.55/2.22	
3.55/2.22	
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(4)
3.55/2.22	Complex Obligation (BEST)
3.55/2.22	
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(5)
3.55/2.22	Obligation:
3.55/2.22	Proved the lower bound n^1 for the following obligation:
3.55/2.22	
3.55/2.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.55/2.22	
3.55/2.22	
3.55/2.22	The TRS R consists of the following rules:
3.55/2.22	
3.55/2.22	   f(s(s(s(s(s(s(s(s(x)))))))), y, y) -> f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
3.55/2.22	   id(s(x)) -> s(id(x))
3.55/2.22	   id(0) -> 0
3.55/2.22	
3.55/2.22	S is empty.
3.55/2.22	Rewrite Strategy: FULL
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(6) LowerBoundPropagationProof (FINISHED)
3.55/2.22	Propagated lower bound.
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(7)
3.55/2.22	BOUNDS(n^1, INF)
3.55/2.22	
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(8)
3.55/2.22	Obligation:
3.55/2.22	Analyzing the following TRS for decreasing loops:
3.55/2.22	
3.55/2.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.55/2.22	
3.55/2.22	
3.55/2.22	The TRS R consists of the following rules:
3.55/2.22	
3.55/2.22	   f(s(s(s(s(s(s(s(s(x)))))))), y, y) -> f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
3.55/2.22	   id(s(x)) -> s(id(x))
3.55/2.22	   id(0) -> 0
3.55/2.22	
3.55/2.22	S is empty.
3.55/2.22	Rewrite Strategy: FULL
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(9) InfiniteLowerBoundProof (FINISHED)
3.55/2.22	The following loop proves infinite runtime complexity:
3.55/2.22	
3.55/2.22	The rewrite sequence
3.55/2.22	
3.55/2.22	f(s(s(s(s(s(s(s(s(x)))))))), y1_2, y1_2) ->^+ f(s(s(s(s(s(s(s(s(id(x))))))))), y1_2, y1_2)
3.55/2.22	
3.55/2.22	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.55/2.22	
3.55/2.22	The pumping substitution is [ ].
3.55/2.22	
3.55/2.22	The result substitution is [x / id(x)].
3.55/2.22	
3.55/2.22	
3.55/2.22	
3.55/2.22	
3.55/2.22	----------------------------------------
3.55/2.22	
3.55/2.22	(10)
3.55/2.22	BOUNDS(INF, INF)
3.55/2.25	EOF