3.04/1.58	WORST_CASE(NON_POLY, ?)
3.04/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.04/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.04/1.59	
3.04/1.59	
3.04/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.04/1.59	
3.04/1.59	(0) CpxTRS
3.04/1.59	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.04/1.59	(2) TRS for Loop Detection
3.04/1.59	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.04/1.59	(4) BEST
3.04/1.59	    (5) proven lower bound
3.04/1.59	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.04/1.59	        (7) BOUNDS(n^1, INF)
3.04/1.59	    (8) TRS for Loop Detection
3.04/1.59	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.04/1.59	        (10) BOUNDS(INF, INF)
3.04/1.59	
3.04/1.59	
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(0)
3.04/1.59	Obligation:
3.04/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.04/1.59	
3.04/1.59	
3.04/1.59	The TRS R consists of the following rules:
3.04/1.59	
3.04/1.59	   from(X) -> cons(X, from(s(X)))
3.04/1.59	   first(0, Z) -> nil
3.04/1.59	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.04/1.59	   sel(0, cons(X, Z)) -> X
3.04/1.59	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.04/1.59	
3.04/1.59	S is empty.
3.04/1.59	Rewrite Strategy: INNERMOST
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.04/1.59	Transformed a relative TRS into a decreasing-loop problem.
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(2)
3.04/1.59	Obligation:
3.04/1.59	Analyzing the following TRS for decreasing loops:
3.04/1.59	
3.04/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.04/1.59	
3.04/1.59	
3.04/1.59	The TRS R consists of the following rules:
3.04/1.59	
3.04/1.59	   from(X) -> cons(X, from(s(X)))
3.04/1.59	   first(0, Z) -> nil
3.04/1.59	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.04/1.59	   sel(0, cons(X, Z)) -> X
3.04/1.59	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.04/1.59	
3.04/1.59	S is empty.
3.04/1.59	Rewrite Strategy: INNERMOST
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.04/1.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.04/1.59	
3.04/1.59	The rewrite sequence
3.04/1.59	
3.04/1.59	first(s(X), cons(Y, Z)) ->^+ cons(Y, first(X, Z))
3.04/1.59	
3.04/1.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.04/1.59	
3.04/1.59	The pumping substitution is [X / s(X), Z / cons(Y, Z)].
3.04/1.59	
3.04/1.59	The result substitution is [ ].
3.04/1.59	
3.04/1.59	
3.04/1.59	
3.04/1.59	
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(4)
3.04/1.59	Complex Obligation (BEST)
3.04/1.59	
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(5)
3.04/1.59	Obligation:
3.04/1.59	Proved the lower bound n^1 for the following obligation:
3.04/1.59	
3.04/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.04/1.59	
3.04/1.59	
3.04/1.59	The TRS R consists of the following rules:
3.04/1.59	
3.04/1.59	   from(X) -> cons(X, from(s(X)))
3.04/1.59	   first(0, Z) -> nil
3.04/1.59	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.04/1.59	   sel(0, cons(X, Z)) -> X
3.04/1.59	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.04/1.59	
3.04/1.59	S is empty.
3.04/1.59	Rewrite Strategy: INNERMOST
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(6) LowerBoundPropagationProof (FINISHED)
3.04/1.59	Propagated lower bound.
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(7)
3.04/1.59	BOUNDS(n^1, INF)
3.04/1.59	
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(8)
3.04/1.59	Obligation:
3.04/1.59	Analyzing the following TRS for decreasing loops:
3.04/1.59	
3.04/1.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.04/1.59	
3.04/1.59	
3.04/1.59	The TRS R consists of the following rules:
3.04/1.59	
3.04/1.59	   from(X) -> cons(X, from(s(X)))
3.04/1.59	   first(0, Z) -> nil
3.04/1.59	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.04/1.59	   sel(0, cons(X, Z)) -> X
3.04/1.59	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.04/1.59	
3.04/1.59	S is empty.
3.04/1.59	Rewrite Strategy: INNERMOST
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(9) InfiniteLowerBoundProof (FINISHED)
3.04/1.59	The following loop proves infinite runtime complexity:
3.04/1.59	
3.04/1.59	The rewrite sequence
3.04/1.59	
3.04/1.59	from(X) ->^+ cons(X, from(s(X)))
3.04/1.59	
3.04/1.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.04/1.59	
3.04/1.59	The pumping substitution is [ ].
3.04/1.59	
3.04/1.59	The result substitution is [X / s(X)].
3.04/1.59	
3.04/1.59	
3.04/1.59	
3.04/1.59	
3.04/1.59	----------------------------------------
3.04/1.59	
3.04/1.59	(10)
3.04/1.59	BOUNDS(INF, INF)
3.29/2.53	EOF