1124.11/291.49	WORST_CASE(Omega(n^1), ?)
1124.11/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1124.11/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1124.11/291.50	
1124.11/291.50	(0) CpxTRS
1124.11/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1124.11/291.50	(2) TRS for Loop Detection
1124.11/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1124.11/291.50	(4) BEST
1124.11/291.50	    (5) proven lower bound
1124.11/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1124.11/291.50	        (7) BOUNDS(n^1, INF)
1124.11/291.50	    (8) TRS for Loop Detection
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(0)
1124.11/291.50	Obligation:
1124.11/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	The TRS R consists of the following rules:
1124.11/291.50	
1124.11/291.50	   a__f(a, b, X) -> a__f(mark(X), X, mark(X))
1124.11/291.50	   a__c -> a
1124.11/291.50	   a__c -> b
1124.11/291.50	   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
1124.11/291.50	   mark(c) -> a__c
1124.11/291.50	   mark(a) -> a
1124.11/291.50	   mark(b) -> b
1124.11/291.50	   a__f(X1, X2, X3) -> f(X1, X2, X3)
1124.11/291.50	   a__c -> c
1124.11/291.50	
1124.11/291.50	S is empty.
1124.11/291.50	Rewrite Strategy: INNERMOST
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1124.11/291.50	Transformed a relative TRS into a decreasing-loop problem.
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(2)
1124.11/291.50	Obligation:
1124.11/291.50	Analyzing the following TRS for decreasing loops:
1124.11/291.50	
1124.11/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	The TRS R consists of the following rules:
1124.11/291.50	
1124.11/291.50	   a__f(a, b, X) -> a__f(mark(X), X, mark(X))
1124.11/291.50	   a__c -> a
1124.11/291.50	   a__c -> b
1124.11/291.50	   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
1124.11/291.50	   mark(c) -> a__c
1124.11/291.50	   mark(a) -> a
1124.11/291.50	   mark(b) -> b
1124.11/291.50	   a__f(X1, X2, X3) -> f(X1, X2, X3)
1124.11/291.50	   a__c -> c
1124.11/291.50	
1124.11/291.50	S is empty.
1124.11/291.50	Rewrite Strategy: INNERMOST
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
1124.11/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1124.11/291.50	
1124.11/291.50	The rewrite sequence
1124.11/291.50	
1124.11/291.50	mark(f(X1, X2, X3)) ->^+ a__f(mark(X1), X2, mark(X3))
1124.11/291.50	
1124.11/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1124.11/291.50	
1124.11/291.50	The pumping substitution is [X1 / f(X1, X2, X3)].
1124.11/291.50	
1124.11/291.50	The result substitution is [ ].
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(4)
1124.11/291.50	Complex Obligation (BEST)
1124.11/291.50	
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(5)
1124.11/291.50	Obligation:
1124.11/291.50	Proved the lower bound n^1 for the following obligation:
1124.11/291.50	
1124.11/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	The TRS R consists of the following rules:
1124.11/291.50	
1124.11/291.50	   a__f(a, b, X) -> a__f(mark(X), X, mark(X))
1124.11/291.50	   a__c -> a
1124.11/291.50	   a__c -> b
1124.11/291.50	   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
1124.11/291.50	   mark(c) -> a__c
1124.11/291.50	   mark(a) -> a
1124.11/291.50	   mark(b) -> b
1124.11/291.50	   a__f(X1, X2, X3) -> f(X1, X2, X3)
1124.11/291.50	   a__c -> c
1124.11/291.50	
1124.11/291.50	S is empty.
1124.11/291.50	Rewrite Strategy: INNERMOST
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(6) LowerBoundPropagationProof (FINISHED)
1124.11/291.50	Propagated lower bound.
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(7)
1124.11/291.50	BOUNDS(n^1, INF)
1124.11/291.50	
1124.11/291.50	----------------------------------------
1124.11/291.50	
1124.11/291.50	(8)
1124.11/291.50	Obligation:
1124.11/291.50	Analyzing the following TRS for decreasing loops:
1124.11/291.50	
1124.11/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1124.11/291.50	
1124.11/291.50	
1124.11/291.50	The TRS R consists of the following rules:
1124.11/291.50	
1124.11/291.50	   a__f(a, b, X) -> a__f(mark(X), X, mark(X))
1124.11/291.50	   a__c -> a
1124.11/291.50	   a__c -> b
1124.11/291.50	   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
1124.11/291.50	   mark(c) -> a__c
1124.11/291.50	   mark(a) -> a
1124.11/291.50	   mark(b) -> b
1124.11/291.50	   a__f(X1, X2, X3) -> f(X1, X2, X3)
1124.11/291.50	   a__c -> c
1124.11/291.50	
1124.11/291.50	S is empty.
1124.11/291.50	Rewrite Strategy: INNERMOST
1124.46/291.62	EOF