3.41/1.68	WORST_CASE(NON_POLY, ?)
3.41/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.69	
3.41/1.69	
3.41/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.69	
3.41/1.69	(0) CpxTRS
3.41/1.69	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.41/1.69	(2) TRS for Loop Detection
3.41/1.69	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.41/1.69	(4) BEST
3.41/1.69	    (5) proven lower bound
3.41/1.69	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.41/1.69	        (7) BOUNDS(n^1, INF)
3.41/1.69	    (8) TRS for Loop Detection
3.41/1.69	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.41/1.69	        (10) BOUNDS(EXP, INF)
3.41/1.69	
3.41/1.69	
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(0)
3.41/1.69	Obligation:
3.41/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.69	
3.41/1.69	
3.41/1.69	The TRS R consists of the following rules:
3.41/1.69	
3.41/1.69	   empty(nil) -> true
3.41/1.69	   empty(cons(x, l)) -> false
3.41/1.69	   head(cons(x, l)) -> x
3.41/1.69	   tail(nil) -> nil
3.41/1.69	   tail(cons(x, l)) -> l
3.41/1.69	   rev(nil) -> nil
3.41/1.69	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
3.41/1.69	   last(x, l) -> if(empty(l), x, l)
3.41/1.69	   if(true, x, l) -> x
3.41/1.69	   if(false, x, l) -> last(head(l), tail(l))
3.41/1.69	   rev2(x, nil) -> nil
3.41/1.69	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	S is empty.
3.41/1.69	Rewrite Strategy: FULL
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.41/1.69	Transformed a relative TRS into a decreasing-loop problem.
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(2)
3.41/1.69	Obligation:
3.41/1.69	Analyzing the following TRS for decreasing loops:
3.41/1.69	
3.41/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.69	
3.41/1.69	
3.41/1.69	The TRS R consists of the following rules:
3.41/1.69	
3.41/1.69	   empty(nil) -> true
3.41/1.69	   empty(cons(x, l)) -> false
3.41/1.69	   head(cons(x, l)) -> x
3.41/1.69	   tail(nil) -> nil
3.41/1.69	   tail(cons(x, l)) -> l
3.41/1.69	   rev(nil) -> nil
3.41/1.69	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
3.41/1.69	   last(x, l) -> if(empty(l), x, l)
3.41/1.69	   if(true, x, l) -> x
3.41/1.69	   if(false, x, l) -> last(head(l), tail(l))
3.41/1.69	   rev2(x, nil) -> nil
3.41/1.69	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	S is empty.
3.41/1.69	Rewrite Strategy: FULL
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.41/1.69	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.41/1.69	
3.41/1.69	The rewrite sequence
3.41/1.69	
3.41/1.69	rev2(x, cons(y, l)) ->^+ rev(cons(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
3.41/1.69	
3.41/1.69	The pumping substitution is [l / cons(y, l)].
3.41/1.69	
3.41/1.69	The result substitution is [x / y].
3.41/1.69	
3.41/1.69	
3.41/1.69	
3.41/1.69	
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(4)
3.41/1.69	Complex Obligation (BEST)
3.41/1.69	
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(5)
3.41/1.69	Obligation:
3.41/1.69	Proved the lower bound n^1 for the following obligation:
3.41/1.69	
3.41/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.69	
3.41/1.69	
3.41/1.69	The TRS R consists of the following rules:
3.41/1.69	
3.41/1.69	   empty(nil) -> true
3.41/1.69	   empty(cons(x, l)) -> false
3.41/1.69	   head(cons(x, l)) -> x
3.41/1.69	   tail(nil) -> nil
3.41/1.69	   tail(cons(x, l)) -> l
3.41/1.69	   rev(nil) -> nil
3.41/1.69	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
3.41/1.69	   last(x, l) -> if(empty(l), x, l)
3.41/1.69	   if(true, x, l) -> x
3.41/1.69	   if(false, x, l) -> last(head(l), tail(l))
3.41/1.69	   rev2(x, nil) -> nil
3.41/1.69	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	S is empty.
3.41/1.69	Rewrite Strategy: FULL
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(6) LowerBoundPropagationProof (FINISHED)
3.41/1.69	Propagated lower bound.
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(7)
3.41/1.69	BOUNDS(n^1, INF)
3.41/1.69	
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(8)
3.41/1.69	Obligation:
3.41/1.69	Analyzing the following TRS for decreasing loops:
3.41/1.69	
3.41/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.41/1.69	
3.41/1.69	
3.41/1.69	The TRS R consists of the following rules:
3.41/1.69	
3.41/1.69	   empty(nil) -> true
3.41/1.69	   empty(cons(x, l)) -> false
3.41/1.69	   head(cons(x, l)) -> x
3.41/1.69	   tail(nil) -> nil
3.41/1.69	   tail(cons(x, l)) -> l
3.41/1.69	   rev(nil) -> nil
3.41/1.69	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
3.41/1.69	   last(x, l) -> if(empty(l), x, l)
3.41/1.69	   if(true, x, l) -> x
3.41/1.69	   if(false, x, l) -> last(head(l), tail(l))
3.41/1.69	   rev2(x, nil) -> nil
3.41/1.69	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	S is empty.
3.41/1.69	Rewrite Strategy: FULL
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(9) DecreasingLoopProof (FINISHED)
3.41/1.69	The following loop(s) give(s) rise to the lower bound EXP:
3.41/1.69	
3.41/1.69	The rewrite sequence
3.41/1.69	
3.41/1.69	rev2(x, cons(y, l)) ->^+ cons(rev1(x, rev2(y, l)), rev2(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
3.41/1.69	
3.41/1.69	The pumping substitution is [l / cons(y, l)].
3.41/1.69	
3.41/1.69	The result substitution is [x / y].
3.41/1.69	
3.41/1.69	
3.41/1.69	
3.41/1.69	The rewrite sequence
3.41/1.69	
3.41/1.69	rev2(x, cons(y, l)) ->^+ cons(rev1(x, rev2(y, l)), rev2(x, rev2(y, l)))
3.41/1.69	
3.41/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
3.41/1.69	
3.41/1.69	The pumping substitution is [l / cons(y, l)].
3.41/1.69	
3.41/1.69	The result substitution is [x / y].
3.41/1.69	
3.41/1.69	
3.41/1.69	
3.41/1.69	
3.41/1.69	----------------------------------------
3.41/1.69	
3.41/1.69	(10)
3.41/1.69	BOUNDS(EXP, INF)
3.68/1.73	EOF