3.12/1.60	WORST_CASE(NON_POLY, ?)
3.12/1.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.12/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.12/1.61	
3.12/1.61	
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.12/1.61	
3.12/1.61	(0) CpxTRS
3.12/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.12/1.61	(2) TRS for Loop Detection
3.12/1.61	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.12/1.61	(4) BEST
3.12/1.61	    (5) proven lower bound
3.12/1.61	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.12/1.61	        (7) BOUNDS(n^1, INF)
3.12/1.61	    (8) TRS for Loop Detection
3.12/1.61	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.12/1.61	        (10) BOUNDS(INF, INF)
3.12/1.61	
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(0)
3.12/1.61	Obligation:
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.12/1.61	
3.12/1.61	
3.12/1.61	The TRS R consists of the following rules:
3.12/1.61	
3.12/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.12/1.61	   sel(0, cons(X, Z)) -> X
3.12/1.61	   first(0, Z) -> nil
3.12/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.12/1.61	   from(X) -> cons(X, from(s(X)))
3.12/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.12/1.61	   sel1(0, cons(X, Z)) -> quote(X)
3.12/1.61	   first1(0, Z) -> nil1
3.12/1.61	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
3.12/1.61	   quote(0) -> 01
3.12/1.61	   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
3.12/1.61	   quote1(nil) -> nil1
3.12/1.61	   quote(s(X)) -> s1(quote(X))
3.12/1.61	   quote(sel(X, Z)) -> sel1(X, Z)
3.12/1.61	   quote1(first(X, Z)) -> first1(X, Z)
3.12/1.61	   unquote(01) -> 0
3.12/1.61	   unquote(s1(X)) -> s(unquote(X))
3.12/1.61	   unquote1(nil1) -> nil
3.12/1.61	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.12/1.61	   fcons(X, Z) -> cons(X, Z)
3.12/1.61	
3.12/1.61	S is empty.
3.12/1.61	Rewrite Strategy: FULL
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.12/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(2)
3.12/1.61	Obligation:
3.12/1.61	Analyzing the following TRS for decreasing loops:
3.12/1.61	
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.12/1.61	
3.12/1.61	
3.12/1.61	The TRS R consists of the following rules:
3.12/1.61	
3.12/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.12/1.61	   sel(0, cons(X, Z)) -> X
3.12/1.61	   first(0, Z) -> nil
3.12/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.12/1.61	   from(X) -> cons(X, from(s(X)))
3.12/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.12/1.61	   sel1(0, cons(X, Z)) -> quote(X)
3.12/1.61	   first1(0, Z) -> nil1
3.12/1.61	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
3.12/1.61	   quote(0) -> 01
3.12/1.61	   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
3.12/1.61	   quote1(nil) -> nil1
3.12/1.61	   quote(s(X)) -> s1(quote(X))
3.12/1.61	   quote(sel(X, Z)) -> sel1(X, Z)
3.12/1.61	   quote1(first(X, Z)) -> first1(X, Z)
3.12/1.61	   unquote(01) -> 0
3.12/1.61	   unquote(s1(X)) -> s(unquote(X))
3.12/1.61	   unquote1(nil1) -> nil
3.12/1.61	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.12/1.61	   fcons(X, Z) -> cons(X, Z)
3.12/1.61	
3.12/1.61	S is empty.
3.12/1.61	Rewrite Strategy: FULL
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.12/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.12/1.61	
3.12/1.61	The rewrite sequence
3.12/1.61	
3.12/1.61	unquote1(cons1(X, Z)) ->^+ fcons(unquote(X), unquote1(Z))
3.12/1.61	
3.12/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.12/1.61	
3.12/1.61	The pumping substitution is [Z / cons1(X, Z)].
3.12/1.61	
3.12/1.61	The result substitution is [ ].
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(4)
3.12/1.61	Complex Obligation (BEST)
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(5)
3.12/1.61	Obligation:
3.12/1.61	Proved the lower bound n^1 for the following obligation:
3.12/1.61	
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.12/1.61	
3.12/1.61	
3.12/1.61	The TRS R consists of the following rules:
3.12/1.61	
3.12/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.12/1.61	   sel(0, cons(X, Z)) -> X
3.12/1.61	   first(0, Z) -> nil
3.12/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.12/1.61	   from(X) -> cons(X, from(s(X)))
3.12/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.12/1.61	   sel1(0, cons(X, Z)) -> quote(X)
3.12/1.61	   first1(0, Z) -> nil1
3.12/1.61	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
3.12/1.61	   quote(0) -> 01
3.12/1.61	   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
3.12/1.61	   quote1(nil) -> nil1
3.12/1.61	   quote(s(X)) -> s1(quote(X))
3.12/1.61	   quote(sel(X, Z)) -> sel1(X, Z)
3.12/1.61	   quote1(first(X, Z)) -> first1(X, Z)
3.12/1.61	   unquote(01) -> 0
3.12/1.61	   unquote(s1(X)) -> s(unquote(X))
3.12/1.61	   unquote1(nil1) -> nil
3.12/1.61	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.12/1.61	   fcons(X, Z) -> cons(X, Z)
3.12/1.61	
3.12/1.61	S is empty.
3.12/1.61	Rewrite Strategy: FULL
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(6) LowerBoundPropagationProof (FINISHED)
3.12/1.61	Propagated lower bound.
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(7)
3.12/1.61	BOUNDS(n^1, INF)
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(8)
3.12/1.61	Obligation:
3.12/1.61	Analyzing the following TRS for decreasing loops:
3.12/1.61	
3.12/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.12/1.61	
3.12/1.61	
3.12/1.61	The TRS R consists of the following rules:
3.12/1.61	
3.12/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.12/1.61	   sel(0, cons(X, Z)) -> X
3.12/1.61	   first(0, Z) -> nil
3.12/1.61	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.12/1.61	   from(X) -> cons(X, from(s(X)))
3.12/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.12/1.61	   sel1(0, cons(X, Z)) -> quote(X)
3.12/1.61	   first1(0, Z) -> nil1
3.12/1.61	   first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
3.12/1.61	   quote(0) -> 01
3.12/1.61	   quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
3.12/1.61	   quote1(nil) -> nil1
3.12/1.61	   quote(s(X)) -> s1(quote(X))
3.12/1.61	   quote(sel(X, Z)) -> sel1(X, Z)
3.12/1.61	   quote1(first(X, Z)) -> first1(X, Z)
3.12/1.61	   unquote(01) -> 0
3.12/1.61	   unquote(s1(X)) -> s(unquote(X))
3.12/1.61	   unquote1(nil1) -> nil
3.12/1.61	   unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
3.12/1.61	   fcons(X, Z) -> cons(X, Z)
3.12/1.61	
3.12/1.61	S is empty.
3.12/1.61	Rewrite Strategy: FULL
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(9) InfiniteLowerBoundProof (FINISHED)
3.12/1.61	The following loop proves infinite runtime complexity:
3.12/1.61	
3.12/1.61	The rewrite sequence
3.12/1.61	
3.12/1.61	from(X) ->^+ cons(X, from(s(X)))
3.12/1.61	
3.12/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.12/1.61	
3.12/1.61	The pumping substitution is [ ].
3.12/1.61	
3.12/1.61	The result substitution is [X / s(X)].
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	
3.12/1.61	----------------------------------------
3.12/1.61	
3.12/1.61	(10)
3.12/1.61	BOUNDS(INF, INF)
3.43/1.64	EOF