6.63/2.46	WORST_CASE(NON_POLY, ?)
6.63/2.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.63/2.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.63/2.47	
6.63/2.47	
6.63/2.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.63/2.47	
6.63/2.47	(0) CpxTRS
6.63/2.47	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
6.63/2.47	(2) CpxTRS
6.63/2.47	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
6.63/2.47	(4) CpxTRS
6.63/2.47	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
6.63/2.47	(6) typed CpxTrs
6.63/2.47	(7) OrderProof [LOWER BOUND(ID), 0 ms]
6.63/2.47	(8) typed CpxTrs
6.63/2.47	(9) RewriteLemmaProof [LOWER BOUND(ID), 214 ms]
6.63/2.47	(10) BEST
6.63/2.47	    (11) proven lower bound
6.63/2.47	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
6.63/2.47	        (13) BOUNDS(n^1, INF)
6.63/2.47	    (14) typed CpxTrs
6.63/2.47	        (15) RewriteLemmaProof [FINISHED, 248 ms]
6.63/2.47	        (16) BOUNDS(EXP, INF)
6.63/2.47	
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(0)
6.63/2.47	Obligation:
6.63/2.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.63/2.47	
6.63/2.47	
6.63/2.47	The TRS R consists of the following rules:
6.63/2.47	
6.63/2.47	   rev(nil) -> nil
6.63/2.47	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	   rev1(0, nil) -> 0
6.63/2.47	   rev1(s(x), nil) -> s(x)
6.63/2.47	   rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	   rev2(x, nil) -> nil
6.63/2.47	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	S is empty.
6.63/2.47	Rewrite Strategy: INNERMOST
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(1) RenamingProof (BOTH BOUNDS(ID, ID))
6.63/2.47	Renamed function symbols to avoid clashes with predefined symbol.
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(2)
6.63/2.47	Obligation:
6.63/2.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.63/2.47	
6.63/2.47	
6.63/2.47	The TRS R consists of the following rules:
6.63/2.47	
6.63/2.47	   rev(nil) -> nil
6.63/2.47	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	   rev1(0', nil) -> 0'
6.63/2.47	   rev1(s(x), nil) -> s(x)
6.63/2.47	   rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	   rev2(x, nil) -> nil
6.63/2.47	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	S is empty.
6.63/2.47	Rewrite Strategy: INNERMOST
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(3) SlicingProof (LOWER BOUND(ID))
6.63/2.47	Sliced the following arguments:
6.63/2.47	s/0
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(4)
6.63/2.47	Obligation:
6.63/2.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
6.63/2.47	
6.63/2.47	
6.63/2.47	The TRS R consists of the following rules:
6.63/2.47	
6.63/2.47	   rev(nil) -> nil
6.63/2.47	   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	   rev1(0', nil) -> 0'
6.63/2.47	   rev1(s, nil) -> s
6.63/2.47	   rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	   rev2(x, nil) -> nil
6.63/2.47	   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	S is empty.
6.63/2.47	Rewrite Strategy: INNERMOST
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
6.63/2.47	Infered types.
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(6)
6.63/2.47	Obligation:
6.63/2.47	Innermost TRS:
6.63/2.47	Rules:
6.63/2.47	rev(nil) -> nil
6.63/2.47	rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	rev1(0', nil) -> 0'
6.63/2.47	rev1(s, nil) -> s
6.63/2.47	rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	rev2(x, nil) -> nil
6.63/2.47	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	Types:
6.63/2.47	rev :: nil:cons -> nil:cons
6.63/2.47	nil :: nil:cons
6.63/2.47	cons :: 0':s -> nil:cons -> nil:cons
6.63/2.47	rev1 :: 0':s -> nil:cons -> 0':s
6.63/2.47	rev2 :: 0':s -> nil:cons -> nil:cons
6.63/2.47	0' :: 0':s
6.63/2.47	s :: 0':s
6.63/2.47	hole_nil:cons1_0 :: nil:cons
6.63/2.47	hole_0':s2_0 :: 0':s
6.63/2.47	gen_nil:cons3_0 :: Nat -> nil:cons
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(7) OrderProof (LOWER BOUND(ID))
6.63/2.47	Heuristically decided to analyse the following defined symbols:
6.63/2.47	rev, rev1, rev2
6.63/2.47	
6.63/2.47	They will be analysed ascendingly in the following order:
6.63/2.47	rev1 < rev
6.63/2.47	rev = rev2
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(8)
6.63/2.47	Obligation:
6.63/2.47	Innermost TRS:
6.63/2.47	Rules:
6.63/2.47	rev(nil) -> nil
6.63/2.47	rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	rev1(0', nil) -> 0'
6.63/2.47	rev1(s, nil) -> s
6.63/2.47	rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	rev2(x, nil) -> nil
6.63/2.47	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	Types:
6.63/2.47	rev :: nil:cons -> nil:cons
6.63/2.47	nil :: nil:cons
6.63/2.47	cons :: 0':s -> nil:cons -> nil:cons
6.63/2.47	rev1 :: 0':s -> nil:cons -> 0':s
6.63/2.47	rev2 :: 0':s -> nil:cons -> nil:cons
6.63/2.47	0' :: 0':s
6.63/2.47	s :: 0':s
6.63/2.47	hole_nil:cons1_0 :: nil:cons
6.63/2.47	hole_0':s2_0 :: 0':s
6.63/2.47	gen_nil:cons3_0 :: Nat -> nil:cons
6.63/2.47	
6.63/2.47	
6.63/2.47	Generator Equations:
6.63/2.47	gen_nil:cons3_0(0) <=> nil
6.63/2.47	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
6.63/2.47	
6.63/2.47	
6.63/2.47	The following defined symbols remain to be analysed:
6.63/2.47	rev1, rev, rev2
6.63/2.47	
6.63/2.47	They will be analysed ascendingly in the following order:
6.63/2.47	rev1 < rev
6.63/2.47	rev = rev2
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(9) RewriteLemmaProof (LOWER BOUND(ID))
6.63/2.47	Proved the following rewrite lemma:
6.63/2.47	rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)
6.63/2.47	
6.63/2.47	Induction Base:
6.63/2.47	rev1(0', gen_nil:cons3_0(0)) ->_R^Omega(1)
6.63/2.47	0'
6.63/2.47	
6.63/2.47	Induction Step:
6.63/2.47	rev1(0', gen_nil:cons3_0(+(n5_0, 1))) ->_R^Omega(1)
6.63/2.47	rev1(0', gen_nil:cons3_0(n5_0)) ->_IH
6.63/2.47	0'
6.63/2.47	
6.63/2.47	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(10)
6.63/2.47	Complex Obligation (BEST)
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(11)
6.63/2.47	Obligation:
6.63/2.47	Proved the lower bound n^1 for the following obligation:
6.63/2.47	
6.63/2.47	Innermost TRS:
6.63/2.47	Rules:
6.63/2.47	rev(nil) -> nil
6.63/2.47	rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	rev1(0', nil) -> 0'
6.63/2.47	rev1(s, nil) -> s
6.63/2.47	rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	rev2(x, nil) -> nil
6.63/2.47	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	Types:
6.63/2.47	rev :: nil:cons -> nil:cons
6.63/2.47	nil :: nil:cons
6.63/2.47	cons :: 0':s -> nil:cons -> nil:cons
6.63/2.47	rev1 :: 0':s -> nil:cons -> 0':s
6.63/2.47	rev2 :: 0':s -> nil:cons -> nil:cons
6.63/2.47	0' :: 0':s
6.63/2.47	s :: 0':s
6.63/2.47	hole_nil:cons1_0 :: nil:cons
6.63/2.47	hole_0':s2_0 :: 0':s
6.63/2.47	gen_nil:cons3_0 :: Nat -> nil:cons
6.63/2.47	
6.63/2.47	
6.63/2.47	Generator Equations:
6.63/2.47	gen_nil:cons3_0(0) <=> nil
6.63/2.47	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
6.63/2.47	
6.63/2.47	
6.63/2.47	The following defined symbols remain to be analysed:
6.63/2.47	rev1, rev, rev2
6.63/2.47	
6.63/2.47	They will be analysed ascendingly in the following order:
6.63/2.47	rev1 < rev
6.63/2.47	rev = rev2
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(12) LowerBoundPropagationProof (FINISHED)
6.63/2.47	Propagated lower bound.
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(13)
6.63/2.47	BOUNDS(n^1, INF)
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(14)
6.63/2.47	Obligation:
6.63/2.47	Innermost TRS:
6.63/2.47	Rules:
6.63/2.47	rev(nil) -> nil
6.63/2.47	rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
6.63/2.47	rev1(0', nil) -> 0'
6.63/2.47	rev1(s, nil) -> s
6.63/2.47	rev1(x, cons(y, l)) -> rev1(y, l)
6.63/2.47	rev2(x, nil) -> nil
6.63/2.47	rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))
6.63/2.47	
6.63/2.47	Types:
6.63/2.47	rev :: nil:cons -> nil:cons
6.63/2.47	nil :: nil:cons
6.63/2.47	cons :: 0':s -> nil:cons -> nil:cons
6.63/2.47	rev1 :: 0':s -> nil:cons -> 0':s
6.63/2.47	rev2 :: 0':s -> nil:cons -> nil:cons
6.63/2.47	0' :: 0':s
6.63/2.47	s :: 0':s
6.63/2.47	hole_nil:cons1_0 :: nil:cons
6.63/2.47	hole_0':s2_0 :: 0':s
6.63/2.47	gen_nil:cons3_0 :: Nat -> nil:cons
6.63/2.47	
6.63/2.47	
6.63/2.47	Lemmas:
6.63/2.47	rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)
6.63/2.47	
6.63/2.47	
6.63/2.47	Generator Equations:
6.63/2.47	gen_nil:cons3_0(0) <=> nil
6.63/2.47	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
6.63/2.47	
6.63/2.47	
6.63/2.47	The following defined symbols remain to be analysed:
6.63/2.47	rev2, rev
6.63/2.47	
6.63/2.47	They will be analysed ascendingly in the following order:
6.63/2.47	rev = rev2
6.63/2.47	
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(15) RewriteLemmaProof (FINISHED)
6.63/2.47	Proved the following rewrite lemma:
6.63/2.47	rev2(0', gen_nil:cons3_0(n203_0)) -> gen_nil:cons3_0(n203_0), rt in Omega(EXP)
6.63/2.47	
6.63/2.47	Induction Base:
6.63/2.47	rev2(0', gen_nil:cons3_0(0)) ->_R^Omega(1)
6.63/2.47	nil
6.63/2.47	
6.63/2.47	Induction Step:
6.63/2.47	rev2(0', gen_nil:cons3_0(+(n203_0, 1))) ->_R^Omega(1)
6.63/2.47	rev(cons(0', rev2(0', gen_nil:cons3_0(n203_0)))) ->_IH
6.63/2.47	rev(cons(0', gen_nil:cons3_0(c204_0))) ->_R^Omega(1)
6.63/2.47	cons(rev1(0', gen_nil:cons3_0(n203_0)), rev2(0', gen_nil:cons3_0(n203_0))) ->_L^Omega(1 + n203_0)
6.63/2.47	cons(0', rev2(0', gen_nil:cons3_0(n203_0))) ->_IH
6.63/2.47	cons(0', gen_nil:cons3_0(c204_0))
6.63/2.47	
6.63/2.47	We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
6.63/2.47	----------------------------------------
6.63/2.47	
6.63/2.47	(16)
6.63/2.47	BOUNDS(EXP, INF)
6.63/2.51	EOF