1115.05/291.53	WORST_CASE(Omega(n^1), ?)
1115.05/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1115.05/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.05/291.54	
1115.05/291.54	(0) CpxTRS
1115.05/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1115.05/291.54	(2) TRS for Loop Detection
1115.05/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1115.05/291.54	(4) BEST
1115.05/291.54	    (5) proven lower bound
1115.05/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1115.05/291.54	        (7) BOUNDS(n^1, INF)
1115.05/291.54	    (8) TRS for Loop Detection
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(0)
1115.05/291.54	Obligation:
1115.05/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	The TRS R consists of the following rules:
1115.05/291.54	
1115.05/291.54	   a__f(X) -> cons(mark(X), f(g(X)))
1115.05/291.54	   a__g(0) -> s(0)
1115.05/291.54	   a__g(s(X)) -> s(s(a__g(mark(X))))
1115.05/291.54	   a__sel(0, cons(X, Y)) -> mark(X)
1115.05/291.54	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1115.05/291.54	   mark(f(X)) -> a__f(mark(X))
1115.05/291.54	   mark(g(X)) -> a__g(mark(X))
1115.05/291.54	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1115.05/291.54	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1115.05/291.54	   mark(0) -> 0
1115.05/291.54	   mark(s(X)) -> s(mark(X))
1115.05/291.54	   a__f(X) -> f(X)
1115.05/291.54	   a__g(X) -> g(X)
1115.05/291.54	   a__sel(X1, X2) -> sel(X1, X2)
1115.05/291.54	
1115.05/291.54	S is empty.
1115.05/291.54	Rewrite Strategy: INNERMOST
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1115.05/291.54	Transformed a relative TRS into a decreasing-loop problem.
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(2)
1115.05/291.54	Obligation:
1115.05/291.54	Analyzing the following TRS for decreasing loops:
1115.05/291.54	
1115.05/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	The TRS R consists of the following rules:
1115.05/291.54	
1115.05/291.54	   a__f(X) -> cons(mark(X), f(g(X)))
1115.05/291.54	   a__g(0) -> s(0)
1115.05/291.54	   a__g(s(X)) -> s(s(a__g(mark(X))))
1115.05/291.54	   a__sel(0, cons(X, Y)) -> mark(X)
1115.05/291.54	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1115.05/291.54	   mark(f(X)) -> a__f(mark(X))
1115.05/291.54	   mark(g(X)) -> a__g(mark(X))
1115.05/291.54	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1115.05/291.54	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1115.05/291.54	   mark(0) -> 0
1115.05/291.54	   mark(s(X)) -> s(mark(X))
1115.05/291.54	   a__f(X) -> f(X)
1115.05/291.54	   a__g(X) -> g(X)
1115.05/291.54	   a__sel(X1, X2) -> sel(X1, X2)
1115.05/291.54	
1115.05/291.54	S is empty.
1115.05/291.54	Rewrite Strategy: INNERMOST
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
1115.05/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1115.05/291.54	
1115.05/291.54	The rewrite sequence
1115.05/291.54	
1115.05/291.54	mark(f(X)) ->^+ a__f(mark(X))
1115.05/291.54	
1115.05/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1115.05/291.54	
1115.05/291.54	The pumping substitution is [X / f(X)].
1115.05/291.54	
1115.05/291.54	The result substitution is [ ].
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(4)
1115.05/291.54	Complex Obligation (BEST)
1115.05/291.54	
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(5)
1115.05/291.54	Obligation:
1115.05/291.54	Proved the lower bound n^1 for the following obligation:
1115.05/291.54	
1115.05/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	The TRS R consists of the following rules:
1115.05/291.54	
1115.05/291.54	   a__f(X) -> cons(mark(X), f(g(X)))
1115.05/291.54	   a__g(0) -> s(0)
1115.05/291.54	   a__g(s(X)) -> s(s(a__g(mark(X))))
1115.05/291.54	   a__sel(0, cons(X, Y)) -> mark(X)
1115.05/291.54	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1115.05/291.54	   mark(f(X)) -> a__f(mark(X))
1115.05/291.54	   mark(g(X)) -> a__g(mark(X))
1115.05/291.54	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1115.05/291.54	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1115.05/291.54	   mark(0) -> 0
1115.05/291.54	   mark(s(X)) -> s(mark(X))
1115.05/291.54	   a__f(X) -> f(X)
1115.05/291.54	   a__g(X) -> g(X)
1115.05/291.54	   a__sel(X1, X2) -> sel(X1, X2)
1115.05/291.54	
1115.05/291.54	S is empty.
1115.05/291.54	Rewrite Strategy: INNERMOST
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(6) LowerBoundPropagationProof (FINISHED)
1115.05/291.54	Propagated lower bound.
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(7)
1115.05/291.54	BOUNDS(n^1, INF)
1115.05/291.54	
1115.05/291.54	----------------------------------------
1115.05/291.54	
1115.05/291.54	(8)
1115.05/291.54	Obligation:
1115.05/291.54	Analyzing the following TRS for decreasing loops:
1115.05/291.54	
1115.05/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.05/291.54	
1115.05/291.54	
1115.05/291.54	The TRS R consists of the following rules:
1115.05/291.54	
1115.05/291.54	   a__f(X) -> cons(mark(X), f(g(X)))
1115.05/291.54	   a__g(0) -> s(0)
1115.05/291.54	   a__g(s(X)) -> s(s(a__g(mark(X))))
1115.05/291.54	   a__sel(0, cons(X, Y)) -> mark(X)
1115.05/291.54	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
1115.05/291.54	   mark(f(X)) -> a__f(mark(X))
1115.05/291.54	   mark(g(X)) -> a__g(mark(X))
1115.05/291.54	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1115.05/291.54	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1115.05/291.54	   mark(0) -> 0
1115.05/291.54	   mark(s(X)) -> s(mark(X))
1115.05/291.54	   a__f(X) -> f(X)
1115.05/291.54	   a__g(X) -> g(X)
1115.05/291.54	   a__sel(X1, X2) -> sel(X1, X2)
1115.05/291.54	
1115.05/291.54	S is empty.
1115.05/291.54	Rewrite Strategy: INNERMOST
1115.25/291.60	EOF