1137.50/291.50	WORST_CASE(Omega(n^1), ?)
1137.60/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1137.60/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.60/291.52	
1137.60/291.52	(0) CpxTRS
1137.60/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1137.60/291.52	(2) TRS for Loop Detection
1137.60/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1137.60/291.52	(4) BEST
1137.60/291.52	    (5) proven lower bound
1137.60/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1137.60/291.52	        (7) BOUNDS(n^1, INF)
1137.60/291.52	    (8) TRS for Loop Detection
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(0)
1137.60/291.52	Obligation:
1137.60/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	The TRS R consists of the following rules:
1137.60/291.52	
1137.60/291.52	   a__fst(0, Z) -> nil
1137.60/291.52	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
1137.60/291.52	   a__from(X) -> cons(mark(X), from(s(X)))
1137.60/291.52	   a__add(0, X) -> mark(X)
1137.60/291.52	   a__add(s(X), Y) -> s(add(X, Y))
1137.60/291.52	   a__len(nil) -> 0
1137.60/291.52	   a__len(cons(X, Z)) -> s(len(Z))
1137.60/291.52	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
1137.60/291.52	   mark(from(X)) -> a__from(mark(X))
1137.60/291.52	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1137.60/291.52	   mark(len(X)) -> a__len(mark(X))
1137.60/291.52	   mark(0) -> 0
1137.60/291.52	   mark(s(X)) -> s(X)
1137.60/291.52	   mark(nil) -> nil
1137.60/291.52	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1137.60/291.52	   a__fst(X1, X2) -> fst(X1, X2)
1137.60/291.52	   a__from(X) -> from(X)
1137.60/291.52	   a__add(X1, X2) -> add(X1, X2)
1137.60/291.52	   a__len(X) -> len(X)
1137.60/291.52	
1137.60/291.52	S is empty.
1137.60/291.52	Rewrite Strategy: INNERMOST
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1137.60/291.52	Transformed a relative TRS into a decreasing-loop problem.
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(2)
1137.60/291.52	Obligation:
1137.60/291.52	Analyzing the following TRS for decreasing loops:
1137.60/291.52	
1137.60/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	The TRS R consists of the following rules:
1137.60/291.52	
1137.60/291.52	   a__fst(0, Z) -> nil
1137.60/291.52	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
1137.60/291.52	   a__from(X) -> cons(mark(X), from(s(X)))
1137.60/291.52	   a__add(0, X) -> mark(X)
1137.60/291.52	   a__add(s(X), Y) -> s(add(X, Y))
1137.60/291.52	   a__len(nil) -> 0
1137.60/291.52	   a__len(cons(X, Z)) -> s(len(Z))
1137.60/291.52	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
1137.60/291.52	   mark(from(X)) -> a__from(mark(X))
1137.60/291.52	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1137.60/291.52	   mark(len(X)) -> a__len(mark(X))
1137.60/291.52	   mark(0) -> 0
1137.60/291.52	   mark(s(X)) -> s(X)
1137.60/291.52	   mark(nil) -> nil
1137.60/291.52	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1137.60/291.52	   a__fst(X1, X2) -> fst(X1, X2)
1137.60/291.52	   a__from(X) -> from(X)
1137.60/291.52	   a__add(X1, X2) -> add(X1, X2)
1137.60/291.52	   a__len(X) -> len(X)
1137.60/291.52	
1137.60/291.52	S is empty.
1137.60/291.52	Rewrite Strategy: INNERMOST
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
1137.60/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1137.60/291.52	
1137.60/291.52	The rewrite sequence
1137.60/291.52	
1137.60/291.52	mark(len(X)) ->^+ a__len(mark(X))
1137.60/291.52	
1137.60/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1137.60/291.52	
1137.60/291.52	The pumping substitution is [X / len(X)].
1137.60/291.52	
1137.60/291.52	The result substitution is [ ].
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(4)
1137.60/291.52	Complex Obligation (BEST)
1137.60/291.52	
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(5)
1137.60/291.52	Obligation:
1137.60/291.52	Proved the lower bound n^1 for the following obligation:
1137.60/291.52	
1137.60/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	The TRS R consists of the following rules:
1137.60/291.52	
1137.60/291.52	   a__fst(0, Z) -> nil
1137.60/291.52	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
1137.60/291.52	   a__from(X) -> cons(mark(X), from(s(X)))
1137.60/291.52	   a__add(0, X) -> mark(X)
1137.60/291.52	   a__add(s(X), Y) -> s(add(X, Y))
1137.60/291.52	   a__len(nil) -> 0
1137.60/291.52	   a__len(cons(X, Z)) -> s(len(Z))
1137.60/291.52	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
1137.60/291.52	   mark(from(X)) -> a__from(mark(X))
1137.60/291.52	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1137.60/291.52	   mark(len(X)) -> a__len(mark(X))
1137.60/291.52	   mark(0) -> 0
1137.60/291.52	   mark(s(X)) -> s(X)
1137.60/291.52	   mark(nil) -> nil
1137.60/291.52	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1137.60/291.52	   a__fst(X1, X2) -> fst(X1, X2)
1137.60/291.52	   a__from(X) -> from(X)
1137.60/291.52	   a__add(X1, X2) -> add(X1, X2)
1137.60/291.52	   a__len(X) -> len(X)
1137.60/291.52	
1137.60/291.52	S is empty.
1137.60/291.52	Rewrite Strategy: INNERMOST
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(6) LowerBoundPropagationProof (FINISHED)
1137.60/291.52	Propagated lower bound.
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(7)
1137.60/291.52	BOUNDS(n^1, INF)
1137.60/291.52	
1137.60/291.52	----------------------------------------
1137.60/291.52	
1137.60/291.52	(8)
1137.60/291.52	Obligation:
1137.60/291.52	Analyzing the following TRS for decreasing loops:
1137.60/291.52	
1137.60/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1137.60/291.52	
1137.60/291.52	
1137.60/291.52	The TRS R consists of the following rules:
1137.60/291.52	
1137.60/291.52	   a__fst(0, Z) -> nil
1137.60/291.52	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
1137.60/291.52	   a__from(X) -> cons(mark(X), from(s(X)))
1137.60/291.52	   a__add(0, X) -> mark(X)
1137.60/291.52	   a__add(s(X), Y) -> s(add(X, Y))
1137.60/291.52	   a__len(nil) -> 0
1137.60/291.52	   a__len(cons(X, Z)) -> s(len(Z))
1137.60/291.52	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
1137.60/291.52	   mark(from(X)) -> a__from(mark(X))
1137.60/291.52	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1137.60/291.52	   mark(len(X)) -> a__len(mark(X))
1137.60/291.52	   mark(0) -> 0
1137.60/291.52	   mark(s(X)) -> s(X)
1137.60/291.52	   mark(nil) -> nil
1137.60/291.52	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1137.60/291.52	   a__fst(X1, X2) -> fst(X1, X2)
1137.60/291.52	   a__from(X) -> from(X)
1137.60/291.52	   a__add(X1, X2) -> add(X1, X2)
1137.60/291.52	   a__len(X) -> len(X)
1137.60/291.52	
1137.60/291.52	S is empty.
1137.60/291.52	Rewrite Strategy: INNERMOST
1137.66/291.60	EOF