1075.13/291.52	WORST_CASE(Omega(n^1), ?)
1075.13/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1075.13/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1075.13/291.53	
1075.13/291.53	(0) CpxTRS
1075.13/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1075.13/291.53	(2) TRS for Loop Detection
1075.13/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1075.13/291.53	(4) BEST
1075.13/291.53	    (5) proven lower bound
1075.13/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1075.13/291.53	        (7) BOUNDS(n^1, INF)
1075.13/291.53	    (8) TRS for Loop Detection
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(0)
1075.13/291.53	Obligation:
1075.13/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	The TRS R consists of the following rules:
1075.13/291.53	
1075.13/291.53	   a__nats -> cons(0, incr(nats))
1075.13/291.53	   a__pairs -> cons(0, incr(odds))
1075.13/291.53	   a__odds -> a__incr(a__pairs)
1075.13/291.53	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1075.13/291.53	   a__head(cons(X, XS)) -> mark(X)
1075.13/291.53	   a__tail(cons(X, XS)) -> mark(XS)
1075.13/291.53	   mark(nats) -> a__nats
1075.13/291.53	   mark(pairs) -> a__pairs
1075.13/291.53	   mark(odds) -> a__odds
1075.13/291.53	   mark(incr(X)) -> a__incr(mark(X))
1075.13/291.53	   mark(head(X)) -> a__head(mark(X))
1075.13/291.53	   mark(tail(X)) -> a__tail(mark(X))
1075.13/291.53	   mark(0) -> 0
1075.13/291.53	   mark(s(X)) -> s(mark(X))
1075.13/291.53	   mark(nil) -> nil
1075.13/291.53	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1075.13/291.53	   a__nats -> nats
1075.13/291.53	   a__pairs -> pairs
1075.13/291.53	   a__odds -> odds
1075.13/291.53	   a__incr(X) -> incr(X)
1075.13/291.53	   a__head(X) -> head(X)
1075.13/291.53	   a__tail(X) -> tail(X)
1075.13/291.53	
1075.13/291.53	S is empty.
1075.13/291.53	Rewrite Strategy: FULL
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1075.13/291.53	Transformed a relative TRS into a decreasing-loop problem.
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(2)
1075.13/291.53	Obligation:
1075.13/291.53	Analyzing the following TRS for decreasing loops:
1075.13/291.53	
1075.13/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	The TRS R consists of the following rules:
1075.13/291.53	
1075.13/291.53	   a__nats -> cons(0, incr(nats))
1075.13/291.53	   a__pairs -> cons(0, incr(odds))
1075.13/291.53	   a__odds -> a__incr(a__pairs)
1075.13/291.53	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1075.13/291.53	   a__head(cons(X, XS)) -> mark(X)
1075.13/291.53	   a__tail(cons(X, XS)) -> mark(XS)
1075.13/291.53	   mark(nats) -> a__nats
1075.13/291.53	   mark(pairs) -> a__pairs
1075.13/291.53	   mark(odds) -> a__odds
1075.13/291.53	   mark(incr(X)) -> a__incr(mark(X))
1075.13/291.53	   mark(head(X)) -> a__head(mark(X))
1075.13/291.53	   mark(tail(X)) -> a__tail(mark(X))
1075.13/291.53	   mark(0) -> 0
1075.13/291.53	   mark(s(X)) -> s(mark(X))
1075.13/291.53	   mark(nil) -> nil
1075.13/291.53	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1075.13/291.53	   a__nats -> nats
1075.13/291.53	   a__pairs -> pairs
1075.13/291.53	   a__odds -> odds
1075.13/291.53	   a__incr(X) -> incr(X)
1075.13/291.53	   a__head(X) -> head(X)
1075.13/291.53	   a__tail(X) -> tail(X)
1075.13/291.53	
1075.13/291.53	S is empty.
1075.13/291.53	Rewrite Strategy: FULL
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
1075.13/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1075.13/291.53	
1075.13/291.53	The rewrite sequence
1075.13/291.53	
1075.13/291.53	mark(incr(X)) ->^+ a__incr(mark(X))
1075.13/291.53	
1075.13/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1075.13/291.53	
1075.13/291.53	The pumping substitution is [X / incr(X)].
1075.13/291.53	
1075.13/291.53	The result substitution is [ ].
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(4)
1075.13/291.53	Complex Obligation (BEST)
1075.13/291.53	
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(5)
1075.13/291.53	Obligation:
1075.13/291.53	Proved the lower bound n^1 for the following obligation:
1075.13/291.53	
1075.13/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	The TRS R consists of the following rules:
1075.13/291.53	
1075.13/291.53	   a__nats -> cons(0, incr(nats))
1075.13/291.53	   a__pairs -> cons(0, incr(odds))
1075.13/291.53	   a__odds -> a__incr(a__pairs)
1075.13/291.53	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1075.13/291.53	   a__head(cons(X, XS)) -> mark(X)
1075.13/291.53	   a__tail(cons(X, XS)) -> mark(XS)
1075.13/291.53	   mark(nats) -> a__nats
1075.13/291.53	   mark(pairs) -> a__pairs
1075.13/291.53	   mark(odds) -> a__odds
1075.13/291.53	   mark(incr(X)) -> a__incr(mark(X))
1075.13/291.53	   mark(head(X)) -> a__head(mark(X))
1075.13/291.53	   mark(tail(X)) -> a__tail(mark(X))
1075.13/291.53	   mark(0) -> 0
1075.13/291.53	   mark(s(X)) -> s(mark(X))
1075.13/291.53	   mark(nil) -> nil
1075.13/291.53	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1075.13/291.53	   a__nats -> nats
1075.13/291.53	   a__pairs -> pairs
1075.13/291.53	   a__odds -> odds
1075.13/291.53	   a__incr(X) -> incr(X)
1075.13/291.53	   a__head(X) -> head(X)
1075.13/291.53	   a__tail(X) -> tail(X)
1075.13/291.53	
1075.13/291.53	S is empty.
1075.13/291.53	Rewrite Strategy: FULL
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(6) LowerBoundPropagationProof (FINISHED)
1075.13/291.53	Propagated lower bound.
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(7)
1075.13/291.53	BOUNDS(n^1, INF)
1075.13/291.53	
1075.13/291.53	----------------------------------------
1075.13/291.53	
1075.13/291.53	(8)
1075.13/291.53	Obligation:
1075.13/291.53	Analyzing the following TRS for decreasing loops:
1075.13/291.53	
1075.13/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1075.13/291.53	
1075.13/291.53	
1075.13/291.53	The TRS R consists of the following rules:
1075.13/291.53	
1075.13/291.53	   a__nats -> cons(0, incr(nats))
1075.13/291.53	   a__pairs -> cons(0, incr(odds))
1075.13/291.53	   a__odds -> a__incr(a__pairs)
1075.13/291.53	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1075.13/291.53	   a__head(cons(X, XS)) -> mark(X)
1075.13/291.53	   a__tail(cons(X, XS)) -> mark(XS)
1075.13/291.53	   mark(nats) -> a__nats
1075.13/291.53	   mark(pairs) -> a__pairs
1075.13/291.53	   mark(odds) -> a__odds
1075.13/291.53	   mark(incr(X)) -> a__incr(mark(X))
1075.13/291.53	   mark(head(X)) -> a__head(mark(X))
1075.13/291.53	   mark(tail(X)) -> a__tail(mark(X))
1075.13/291.53	   mark(0) -> 0
1075.13/291.53	   mark(s(X)) -> s(mark(X))
1075.13/291.53	   mark(nil) -> nil
1075.13/291.53	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1075.13/291.53	   a__nats -> nats
1075.13/291.53	   a__pairs -> pairs
1075.13/291.53	   a__odds -> odds
1075.13/291.53	   a__incr(X) -> incr(X)
1075.13/291.53	   a__head(X) -> head(X)
1075.13/291.53	   a__tail(X) -> tail(X)
1075.13/291.53	
1075.13/291.53	S is empty.
1075.13/291.53	Rewrite Strategy: FULL
1075.54/291.59	EOF