3.27/1.59	WORST_CASE(NON_POLY, ?)
3.38/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.38/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.38/1.60	
3.38/1.60	
3.38/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.60	
3.38/1.60	(0) CpxTRS
3.38/1.60	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.38/1.60	(2) TRS for Loop Detection
3.38/1.60	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.38/1.60	(4) BEST
3.38/1.60	    (5) proven lower bound
3.38/1.60	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.38/1.60	        (7) BOUNDS(n^1, INF)
3.38/1.60	    (8) TRS for Loop Detection
3.38/1.60	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.38/1.60	        (10) BOUNDS(EXP, INF)
3.38/1.60	
3.38/1.60	
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(0)
3.38/1.60	Obligation:
3.38/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.60	
3.38/1.60	
3.38/1.60	The TRS R consists of the following rules:
3.38/1.60	
3.38/1.60	   rev1(0, nil) -> 0
3.38/1.60	   rev1(s(X), nil) -> s(X)
3.38/1.60	   rev1(X, cons(Y, L)) -> rev1(Y, L)
3.38/1.60	   rev(nil) -> nil
3.38/1.60	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
3.38/1.60	   rev2(X, nil) -> nil
3.38/1.60	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	S is empty.
3.38/1.60	Rewrite Strategy: FULL
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.38/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(2)
3.38/1.60	Obligation:
3.38/1.60	Analyzing the following TRS for decreasing loops:
3.38/1.60	
3.38/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.60	
3.38/1.60	
3.38/1.60	The TRS R consists of the following rules:
3.38/1.60	
3.38/1.60	   rev1(0, nil) -> 0
3.38/1.60	   rev1(s(X), nil) -> s(X)
3.38/1.60	   rev1(X, cons(Y, L)) -> rev1(Y, L)
3.38/1.60	   rev(nil) -> nil
3.38/1.60	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
3.38/1.60	   rev2(X, nil) -> nil
3.38/1.60	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	S is empty.
3.38/1.60	Rewrite Strategy: FULL
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.38/1.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.38/1.60	
3.38/1.60	The rewrite sequence
3.38/1.60	
3.38/1.60	rev2(X, cons(Y, L)) ->^+ rev(cons(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
3.38/1.60	
3.38/1.60	The pumping substitution is [L / cons(Y, L)].
3.38/1.60	
3.38/1.60	The result substitution is [X / Y].
3.38/1.60	
3.38/1.60	
3.38/1.60	
3.38/1.60	
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(4)
3.38/1.60	Complex Obligation (BEST)
3.38/1.60	
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(5)
3.38/1.60	Obligation:
3.38/1.60	Proved the lower bound n^1 for the following obligation:
3.38/1.60	
3.38/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.60	
3.38/1.60	
3.38/1.60	The TRS R consists of the following rules:
3.38/1.60	
3.38/1.60	   rev1(0, nil) -> 0
3.38/1.60	   rev1(s(X), nil) -> s(X)
3.38/1.60	   rev1(X, cons(Y, L)) -> rev1(Y, L)
3.38/1.60	   rev(nil) -> nil
3.38/1.60	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
3.38/1.60	   rev2(X, nil) -> nil
3.38/1.60	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	S is empty.
3.38/1.60	Rewrite Strategy: FULL
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(6) LowerBoundPropagationProof (FINISHED)
3.38/1.60	Propagated lower bound.
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(7)
3.38/1.60	BOUNDS(n^1, INF)
3.38/1.60	
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(8)
3.38/1.60	Obligation:
3.38/1.60	Analyzing the following TRS for decreasing loops:
3.38/1.60	
3.38/1.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.38/1.60	
3.38/1.60	
3.38/1.60	The TRS R consists of the following rules:
3.38/1.60	
3.38/1.60	   rev1(0, nil) -> 0
3.38/1.60	   rev1(s(X), nil) -> s(X)
3.38/1.60	   rev1(X, cons(Y, L)) -> rev1(Y, L)
3.38/1.60	   rev(nil) -> nil
3.38/1.60	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
3.38/1.60	   rev2(X, nil) -> nil
3.38/1.60	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	S is empty.
3.38/1.60	Rewrite Strategy: FULL
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(9) DecreasingLoopProof (FINISHED)
3.38/1.60	The following loop(s) give(s) rise to the lower bound EXP:
3.38/1.60	
3.38/1.60	The rewrite sequence
3.38/1.60	
3.38/1.60	rev2(X, cons(Y, L)) ->^+ cons(rev1(X, rev(rev2(Y, L))), rev2(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
3.38/1.60	
3.38/1.60	The pumping substitution is [L / cons(Y, L)].
3.38/1.60	
3.38/1.60	The result substitution is [X / Y].
3.38/1.60	
3.38/1.60	
3.38/1.60	
3.38/1.60	The rewrite sequence
3.38/1.60	
3.38/1.60	rev2(X, cons(Y, L)) ->^+ cons(rev1(X, rev(rev2(Y, L))), rev2(X, rev(rev2(Y, L))))
3.38/1.60	
3.38/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,0].
3.38/1.60	
3.38/1.60	The pumping substitution is [L / cons(Y, L)].
3.38/1.60	
3.38/1.60	The result substitution is [X / Y].
3.38/1.60	
3.38/1.60	
3.38/1.60	
3.38/1.60	
3.38/1.60	----------------------------------------
3.38/1.60	
3.38/1.60	(10)
3.38/1.60	BOUNDS(EXP, INF)
3.38/1.62	EOF