908.95/291.52	WORST_CASE(Omega(n^1), ?)
908.95/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
908.95/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
908.95/291.53	
908.95/291.53	
908.95/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
908.95/291.53	
908.95/291.53	(0) CpxTRS
908.95/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
908.95/291.53	(2) TRS for Loop Detection
908.95/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
908.95/291.53	(4) BEST
908.95/291.53	    (5) proven lower bound
908.95/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
908.95/291.53	        (7) BOUNDS(n^1, INF)
908.95/291.53	    (8) TRS for Loop Detection
908.95/291.53	
908.95/291.53	
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(0)
908.95/291.53	Obligation:
908.95/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
908.95/291.53	
908.95/291.53	
908.95/291.53	The TRS R consists of the following rules:
908.95/291.53	
908.95/291.53	   a__f(a, b, X) -> a__f(X, X, mark(X))
908.95/291.53	   a__c -> a
908.95/291.53	   a__c -> b
908.95/291.53	   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
908.95/291.53	   mark(c) -> a__c
908.95/291.53	   mark(a) -> a
908.95/291.53	   mark(b) -> b
908.95/291.53	   a__f(X1, X2, X3) -> f(X1, X2, X3)
908.95/291.53	   a__c -> c
908.95/291.53	
908.95/291.53	S is empty.
908.95/291.53	Rewrite Strategy: INNERMOST
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
908.95/291.53	Transformed a relative TRS into a decreasing-loop problem.
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(2)
908.95/291.53	Obligation:
908.95/291.53	Analyzing the following TRS for decreasing loops:
908.95/291.53	
908.95/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
908.95/291.53	
908.95/291.53	
908.95/291.53	The TRS R consists of the following rules:
908.95/291.53	
908.95/291.53	   a__f(a, b, X) -> a__f(X, X, mark(X))
908.95/291.53	   a__c -> a
908.95/291.53	   a__c -> b
908.95/291.53	   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
908.95/291.53	   mark(c) -> a__c
908.95/291.53	   mark(a) -> a
908.95/291.53	   mark(b) -> b
908.95/291.53	   a__f(X1, X2, X3) -> f(X1, X2, X3)
908.95/291.53	   a__c -> c
908.95/291.53	
908.95/291.53	S is empty.
908.95/291.53	Rewrite Strategy: INNERMOST
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
908.95/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
908.95/291.53	
908.95/291.53	The rewrite sequence
908.95/291.53	
908.95/291.53	mark(f(X1, X2, X3)) ->^+ a__f(X1, X2, mark(X3))
908.95/291.53	
908.95/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [2].
908.95/291.53	
908.95/291.53	The pumping substitution is [X3 / f(X1, X2, X3)].
908.95/291.53	
908.95/291.53	The result substitution is [ ].
908.95/291.53	
908.95/291.53	
908.95/291.53	
908.95/291.53	
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(4)
908.95/291.53	Complex Obligation (BEST)
908.95/291.53	
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(5)
908.95/291.53	Obligation:
908.95/291.53	Proved the lower bound n^1 for the following obligation:
908.95/291.53	
908.95/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
908.95/291.53	
908.95/291.53	
908.95/291.53	The TRS R consists of the following rules:
908.95/291.53	
908.95/291.53	   a__f(a, b, X) -> a__f(X, X, mark(X))
908.95/291.53	   a__c -> a
908.95/291.53	   a__c -> b
908.95/291.53	   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
908.95/291.53	   mark(c) -> a__c
908.95/291.53	   mark(a) -> a
908.95/291.53	   mark(b) -> b
908.95/291.53	   a__f(X1, X2, X3) -> f(X1, X2, X3)
908.95/291.53	   a__c -> c
908.95/291.53	
908.95/291.53	S is empty.
908.95/291.53	Rewrite Strategy: INNERMOST
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(6) LowerBoundPropagationProof (FINISHED)
908.95/291.53	Propagated lower bound.
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(7)
908.95/291.53	BOUNDS(n^1, INF)
908.95/291.53	
908.95/291.53	----------------------------------------
908.95/291.53	
908.95/291.53	(8)
908.95/291.53	Obligation:
908.95/291.53	Analyzing the following TRS for decreasing loops:
908.95/291.53	
908.95/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
908.95/291.53	
908.95/291.53	
908.95/291.53	The TRS R consists of the following rules:
908.95/291.53	
908.95/291.53	   a__f(a, b, X) -> a__f(X, X, mark(X))
908.95/291.53	   a__c -> a
908.95/291.53	   a__c -> b
908.95/291.53	   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
908.95/291.53	   mark(c) -> a__c
908.95/291.53	   mark(a) -> a
908.95/291.53	   mark(b) -> b
908.95/291.53	   a__f(X1, X2, X3) -> f(X1, X2, X3)
908.95/291.53	   a__c -> c
908.95/291.53	
908.95/291.53	S is empty.
908.95/291.53	Rewrite Strategy: INNERMOST
909.15/291.56	EOF