3.13/1.60	WORST_CASE(NON_POLY, ?)
3.13/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.13/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.13/1.60	
3.13/1.60	
3.13/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.13/1.60	
3.13/1.60	(0) CpxTRS
3.13/1.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.13/1.60	(2) TRS for Loop Detection
3.13/1.60	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.13/1.60	(4) BEST
3.13/1.60	    (5) proven lower bound
3.13/1.60	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.13/1.60	        (7) BOUNDS(n^1, INF)
3.13/1.60	    (8) TRS for Loop Detection
3.13/1.60	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.13/1.60	        (10) BOUNDS(INF, INF)
3.13/1.60	
3.13/1.60	
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(0)
3.13/1.60	Obligation:
3.13/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.13/1.60	
3.13/1.60	
3.13/1.60	The TRS R consists of the following rules:
3.13/1.60	
3.13/1.60	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.13/1.60	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.13/1.60	   add(0, X) -> X
3.13/1.60	   add(s(X), Y) -> s(add(X, Y))
3.13/1.60	   sel(0, cons(X, XS)) -> X
3.13/1.60	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.13/1.60	
3.13/1.60	S is empty.
3.13/1.60	Rewrite Strategy: FULL
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.13/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(2)
3.13/1.60	Obligation:
3.13/1.60	Analyzing the following TRS for decreasing loops:
3.13/1.60	
3.13/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.13/1.60	
3.13/1.60	
3.13/1.60	The TRS R consists of the following rules:
3.13/1.60	
3.13/1.60	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.13/1.60	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.13/1.60	   add(0, X) -> X
3.13/1.60	   add(s(X), Y) -> s(add(X, Y))
3.13/1.60	   sel(0, cons(X, XS)) -> X
3.13/1.60	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.13/1.60	
3.13/1.60	S is empty.
3.13/1.60	Rewrite Strategy: FULL
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.13/1.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.13/1.60	
3.13/1.60	The rewrite sequence
3.13/1.60	
3.13/1.60	add(s(X), Y) ->^+ s(add(X, Y))
3.13/1.60	
3.13/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.13/1.60	
3.13/1.60	The pumping substitution is [X / s(X)].
3.13/1.60	
3.13/1.60	The result substitution is [ ].
3.13/1.60	
3.13/1.60	
3.13/1.60	
3.13/1.60	
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(4)
3.13/1.60	Complex Obligation (BEST)
3.13/1.60	
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(5)
3.13/1.60	Obligation:
3.13/1.60	Proved the lower bound n^1 for the following obligation:
3.13/1.60	
3.13/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.13/1.60	
3.13/1.60	
3.13/1.60	The TRS R consists of the following rules:
3.13/1.60	
3.13/1.60	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.13/1.60	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.13/1.60	   add(0, X) -> X
3.13/1.60	   add(s(X), Y) -> s(add(X, Y))
3.13/1.60	   sel(0, cons(X, XS)) -> X
3.13/1.60	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.13/1.60	
3.13/1.60	S is empty.
3.13/1.60	Rewrite Strategy: FULL
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(6) LowerBoundPropagationProof (FINISHED)
3.13/1.60	Propagated lower bound.
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(7)
3.13/1.60	BOUNDS(n^1, INF)
3.13/1.60	
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(8)
3.13/1.60	Obligation:
3.13/1.60	Analyzing the following TRS for decreasing loops:
3.13/1.60	
3.13/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.13/1.60	
3.13/1.60	
3.13/1.60	The TRS R consists of the following rules:
3.13/1.60	
3.13/1.60	   fib(N) -> sel(N, fib1(s(0), s(0)))
3.13/1.60	   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
3.13/1.60	   add(0, X) -> X
3.13/1.60	   add(s(X), Y) -> s(add(X, Y))
3.13/1.60	   sel(0, cons(X, XS)) -> X
3.13/1.60	   sel(s(N), cons(X, XS)) -> sel(N, XS)
3.13/1.60	
3.13/1.60	S is empty.
3.13/1.60	Rewrite Strategy: FULL
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(9) InfiniteLowerBoundProof (FINISHED)
3.13/1.60	The following loop proves infinite runtime complexity:
3.13/1.60	
3.13/1.60	The rewrite sequence
3.13/1.60	
3.13/1.60	fib1(X, Y) ->^+ cons(X, fib1(Y, add(X, Y)))
3.13/1.60	
3.13/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.13/1.60	
3.13/1.60	The pumping substitution is [ ].
3.13/1.60	
3.13/1.60	The result substitution is [X / Y, Y / add(X, Y)].
3.13/1.60	
3.13/1.60	
3.13/1.60	
3.13/1.60	
3.13/1.60	----------------------------------------
3.13/1.60	
3.13/1.60	(10)
3.13/1.60	BOUNDS(INF, INF)
3.41/1.63	EOF