1101.68/291.53	WORST_CASE(Omega(n^1), ?)
1103.77/291.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1103.77/291.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1103.77/291.97	
1103.77/291.97	(0) CpxTRS
1103.77/291.97	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1103.77/291.97	(2) TRS for Loop Detection
1103.77/291.97	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1103.77/291.97	(4) BEST
1103.77/291.97	    (5) proven lower bound
1103.77/291.97	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1103.77/291.97	        (7) BOUNDS(n^1, INF)
1103.77/291.97	    (8) TRS for Loop Detection
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(0)
1103.77/291.97	Obligation:
1103.77/291.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	The TRS R consists of the following rules:
1103.77/291.97	
1103.77/291.97	   a__f(0) -> cons(0, f(s(0)))
1103.77/291.97	   a__f(s(0)) -> a__f(a__p(s(0)))
1103.77/291.97	   a__p(s(X)) -> mark(X)
1103.77/291.97	   mark(f(X)) -> a__f(mark(X))
1103.77/291.97	   mark(p(X)) -> a__p(mark(X))
1103.77/291.97	   mark(0) -> 0
1103.77/291.97	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1103.77/291.97	   mark(s(X)) -> s(mark(X))
1103.77/291.97	   a__f(X) -> f(X)
1103.77/291.97	   a__p(X) -> p(X)
1103.77/291.97	
1103.77/291.97	S is empty.
1103.77/291.97	Rewrite Strategy: FULL
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1103.77/291.97	Transformed a relative TRS into a decreasing-loop problem.
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(2)
1103.77/291.97	Obligation:
1103.77/291.97	Analyzing the following TRS for decreasing loops:
1103.77/291.97	
1103.77/291.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	The TRS R consists of the following rules:
1103.77/291.97	
1103.77/291.97	   a__f(0) -> cons(0, f(s(0)))
1103.77/291.97	   a__f(s(0)) -> a__f(a__p(s(0)))
1103.77/291.97	   a__p(s(X)) -> mark(X)
1103.77/291.97	   mark(f(X)) -> a__f(mark(X))
1103.77/291.97	   mark(p(X)) -> a__p(mark(X))
1103.77/291.97	   mark(0) -> 0
1103.77/291.97	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1103.77/291.97	   mark(s(X)) -> s(mark(X))
1103.77/291.97	   a__f(X) -> f(X)
1103.77/291.97	   a__p(X) -> p(X)
1103.77/291.97	
1103.77/291.97	S is empty.
1103.77/291.97	Rewrite Strategy: FULL
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(3) DecreasingLoopProof (LOWER BOUND(ID))
1103.77/291.97	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1103.77/291.97	
1103.77/291.97	The rewrite sequence
1103.77/291.97	
1103.77/291.97	mark(f(X)) ->^+ a__f(mark(X))
1103.77/291.97	
1103.77/291.97	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1103.77/291.97	
1103.77/291.97	The pumping substitution is [X / f(X)].
1103.77/291.97	
1103.77/291.97	The result substitution is [ ].
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(4)
1103.77/291.97	Complex Obligation (BEST)
1103.77/291.97	
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(5)
1103.77/291.97	Obligation:
1103.77/291.97	Proved the lower bound n^1 for the following obligation:
1103.77/291.97	
1103.77/291.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	The TRS R consists of the following rules:
1103.77/291.97	
1103.77/291.97	   a__f(0) -> cons(0, f(s(0)))
1103.77/291.97	   a__f(s(0)) -> a__f(a__p(s(0)))
1103.77/291.97	   a__p(s(X)) -> mark(X)
1103.77/291.97	   mark(f(X)) -> a__f(mark(X))
1103.77/291.97	   mark(p(X)) -> a__p(mark(X))
1103.77/291.97	   mark(0) -> 0
1103.77/291.97	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1103.77/291.97	   mark(s(X)) -> s(mark(X))
1103.77/291.97	   a__f(X) -> f(X)
1103.77/291.97	   a__p(X) -> p(X)
1103.77/291.97	
1103.77/291.97	S is empty.
1103.77/291.97	Rewrite Strategy: FULL
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(6) LowerBoundPropagationProof (FINISHED)
1103.77/291.97	Propagated lower bound.
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(7)
1103.77/291.97	BOUNDS(n^1, INF)
1103.77/291.97	
1103.77/291.97	----------------------------------------
1103.77/291.97	
1103.77/291.97	(8)
1103.77/291.97	Obligation:
1103.77/291.97	Analyzing the following TRS for decreasing loops:
1103.77/291.97	
1103.77/291.97	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1103.77/291.97	
1103.77/291.97	
1103.77/291.97	The TRS R consists of the following rules:
1103.77/291.97	
1103.77/291.97	   a__f(0) -> cons(0, f(s(0)))
1103.77/291.97	   a__f(s(0)) -> a__f(a__p(s(0)))
1103.77/291.97	   a__p(s(X)) -> mark(X)
1103.77/291.97	   mark(f(X)) -> a__f(mark(X))
1103.77/291.97	   mark(p(X)) -> a__p(mark(X))
1103.77/291.97	   mark(0) -> 0
1103.77/291.97	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1103.77/291.97	   mark(s(X)) -> s(mark(X))
1103.77/291.97	   a__f(X) -> f(X)
1103.77/291.97	   a__p(X) -> p(X)
1103.77/291.97	
1103.77/291.97	S is empty.
1103.77/291.97	Rewrite Strategy: FULL
1103.85/292.03	EOF