3.27/1.59	WORST_CASE(NON_POLY, ?)
3.27/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.27/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.27/1.61	
3.27/1.61	
3.27/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.27/1.61	
3.27/1.61	(0) CpxTRS
3.27/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.27/1.61	(2) TRS for Loop Detection
3.27/1.61	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.27/1.61	(4) BEST
3.27/1.61	    (5) proven lower bound
3.27/1.61	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.27/1.61	        (7) BOUNDS(n^1, INF)
3.27/1.61	    (8) TRS for Loop Detection
3.27/1.61	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.27/1.61	        (10) BOUNDS(INF, INF)
3.27/1.61	
3.27/1.61	
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(0)
3.27/1.61	Obligation:
3.27/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.27/1.61	
3.27/1.61	
3.27/1.61	The TRS R consists of the following rules:
3.27/1.61	
3.27/1.61	   dbl(0) -> 0
3.27/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.27/1.61	   dbls(nil) -> nil
3.27/1.61	   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
3.27/1.61	   sel(0, cons(X, Y)) -> X
3.27/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.27/1.61	   indx(nil, X) -> nil
3.27/1.61	   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
3.27/1.61	   from(X) -> cons(X, from(s(X)))
3.27/1.61	   dbl1(0) -> 01
3.27/1.61	   dbl1(s(X)) -> s1(s1(dbl1(X)))
3.27/1.61	   sel1(0, cons(X, Y)) -> X
3.27/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.27/1.61	   quote(0) -> 01
3.27/1.61	   quote(s(X)) -> s1(quote(X))
3.27/1.61	   quote(dbl(X)) -> dbl1(X)
3.27/1.61	   quote(sel(X, Y)) -> sel1(X, Y)
3.27/1.61	
3.27/1.61	S is empty.
3.27/1.61	Rewrite Strategy: FULL
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.27/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(2)
3.27/1.61	Obligation:
3.27/1.61	Analyzing the following TRS for decreasing loops:
3.27/1.61	
3.27/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.27/1.61	
3.27/1.61	
3.27/1.61	The TRS R consists of the following rules:
3.27/1.61	
3.27/1.61	   dbl(0) -> 0
3.27/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.27/1.61	   dbls(nil) -> nil
3.27/1.61	   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
3.27/1.61	   sel(0, cons(X, Y)) -> X
3.27/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.27/1.61	   indx(nil, X) -> nil
3.27/1.61	   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
3.27/1.61	   from(X) -> cons(X, from(s(X)))
3.27/1.61	   dbl1(0) -> 01
3.27/1.61	   dbl1(s(X)) -> s1(s1(dbl1(X)))
3.27/1.61	   sel1(0, cons(X, Y)) -> X
3.27/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.27/1.61	   quote(0) -> 01
3.27/1.61	   quote(s(X)) -> s1(quote(X))
3.27/1.61	   quote(dbl(X)) -> dbl1(X)
3.27/1.61	   quote(sel(X, Y)) -> sel1(X, Y)
3.27/1.61	
3.27/1.61	S is empty.
3.27/1.61	Rewrite Strategy: FULL
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.27/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.27/1.61	
3.27/1.61	The rewrite sequence
3.27/1.61	
3.27/1.61	dbls(cons(X, Y)) ->^+ cons(dbl(X), dbls(Y))
3.27/1.61	
3.27/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.27/1.61	
3.27/1.61	The pumping substitution is [Y / cons(X, Y)].
3.27/1.61	
3.27/1.61	The result substitution is [ ].
3.27/1.61	
3.27/1.61	
3.27/1.61	
3.27/1.61	
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(4)
3.27/1.61	Complex Obligation (BEST)
3.27/1.61	
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(5)
3.27/1.61	Obligation:
3.27/1.61	Proved the lower bound n^1 for the following obligation:
3.27/1.61	
3.27/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.27/1.61	
3.27/1.61	
3.27/1.61	The TRS R consists of the following rules:
3.27/1.61	
3.27/1.61	   dbl(0) -> 0
3.27/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.27/1.61	   dbls(nil) -> nil
3.27/1.61	   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
3.27/1.61	   sel(0, cons(X, Y)) -> X
3.27/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.27/1.61	   indx(nil, X) -> nil
3.27/1.61	   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
3.27/1.61	   from(X) -> cons(X, from(s(X)))
3.27/1.61	   dbl1(0) -> 01
3.27/1.61	   dbl1(s(X)) -> s1(s1(dbl1(X)))
3.27/1.61	   sel1(0, cons(X, Y)) -> X
3.27/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.27/1.61	   quote(0) -> 01
3.27/1.61	   quote(s(X)) -> s1(quote(X))
3.27/1.61	   quote(dbl(X)) -> dbl1(X)
3.27/1.61	   quote(sel(X, Y)) -> sel1(X, Y)
3.27/1.61	
3.27/1.61	S is empty.
3.27/1.61	Rewrite Strategy: FULL
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(6) LowerBoundPropagationProof (FINISHED)
3.27/1.61	Propagated lower bound.
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(7)
3.27/1.61	BOUNDS(n^1, INF)
3.27/1.61	
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(8)
3.27/1.61	Obligation:
3.27/1.61	Analyzing the following TRS for decreasing loops:
3.27/1.61	
3.27/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.27/1.61	
3.27/1.61	
3.27/1.61	The TRS R consists of the following rules:
3.27/1.61	
3.27/1.61	   dbl(0) -> 0
3.27/1.61	   dbl(s(X)) -> s(s(dbl(X)))
3.27/1.61	   dbls(nil) -> nil
3.27/1.61	   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
3.27/1.61	   sel(0, cons(X, Y)) -> X
3.27/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.27/1.61	   indx(nil, X) -> nil
3.27/1.61	   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
3.27/1.61	   from(X) -> cons(X, from(s(X)))
3.27/1.61	   dbl1(0) -> 01
3.27/1.61	   dbl1(s(X)) -> s1(s1(dbl1(X)))
3.27/1.61	   sel1(0, cons(X, Y)) -> X
3.27/1.61	   sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
3.27/1.61	   quote(0) -> 01
3.27/1.61	   quote(s(X)) -> s1(quote(X))
3.27/1.61	   quote(dbl(X)) -> dbl1(X)
3.27/1.61	   quote(sel(X, Y)) -> sel1(X, Y)
3.27/1.61	
3.27/1.61	S is empty.
3.27/1.61	Rewrite Strategy: FULL
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(9) InfiniteLowerBoundProof (FINISHED)
3.27/1.61	The following loop proves infinite runtime complexity:
3.27/1.61	
3.27/1.61	The rewrite sequence
3.27/1.61	
3.27/1.61	from(X) ->^+ cons(X, from(s(X)))
3.27/1.61	
3.27/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.27/1.61	
3.27/1.61	The pumping substitution is [ ].
3.27/1.61	
3.27/1.61	The result substitution is [X / s(X)].
3.27/1.61	
3.27/1.61	
3.27/1.61	
3.27/1.61	
3.27/1.61	----------------------------------------
3.27/1.61	
3.27/1.61	(10)
3.27/1.61	BOUNDS(INF, INF)
3.39/1.66	EOF