1125.83/292.11	WORST_CASE(Omega(n^1), ?)
1126.11/292.15	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1126.11/292.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1126.11/292.15	
1126.11/292.15	
1126.11/292.15	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.11/292.15	
1126.11/292.15	(0) CpxTRS
1126.11/292.15	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1126.11/292.15	(2) CpxTRS
1126.11/292.15	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
1126.11/292.15	(4) CpxTRS
1126.11/292.15	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1126.11/292.15	(6) typed CpxTrs
1126.11/292.15	(7) OrderProof [LOWER BOUND(ID), 0 ms]
1126.11/292.15	(8) typed CpxTrs
1126.11/292.15	(9) RewriteLemmaProof [LOWER BOUND(ID), 225 ms]
1126.11/292.15	(10) BEST
1126.11/292.15	    (11) proven lower bound
1126.11/292.15	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
1126.11/292.15	        (13) BOUNDS(n^1, INF)
1126.11/292.15	    (14) typed CpxTrs
1126.11/292.15	        (15) RewriteLemmaProof [LOWER BOUND(ID), 578 ms]
1126.11/292.15	        (16) typed CpxTrs
1126.11/292.15	
1126.11/292.15	
1126.11/292.15	----------------------------------------
1126.11/292.15	
1126.11/292.15	(0)
1126.11/292.15	Obligation:
1126.11/292.15	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.11/292.15	
1126.11/292.15	
1126.11/292.15	The TRS R consists of the following rules:
1126.11/292.15	
1126.11/292.15	   rev1(0, nil) -> 0
1126.11/292.15	   rev1(s(X), nil) -> s(X)
1126.11/292.16	   rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	   rev(nil) -> nil
1126.11/292.16	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	   rev2(X, nil) -> nil
1126.11/292.16	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	S is empty.
1126.11/292.16	Rewrite Strategy: INNERMOST
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1126.11/292.16	Renamed function symbols to avoid clashes with predefined symbol.
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(2)
1126.11/292.16	Obligation:
1126.11/292.16	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	The TRS R consists of the following rules:
1126.11/292.16	
1126.11/292.16	   rev1(0', nil) -> 0'
1126.11/292.16	   rev1(s(X), nil) -> s(X)
1126.11/292.16	   rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	   rev(nil) -> nil
1126.11/292.16	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	   rev2(X, nil) -> nil
1126.11/292.16	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	S is empty.
1126.11/292.16	Rewrite Strategy: INNERMOST
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(3) SlicingProof (LOWER BOUND(ID))
1126.11/292.16	Sliced the following arguments:
1126.11/292.16	s/0
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(4)
1126.11/292.16	Obligation:
1126.11/292.16	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	The TRS R consists of the following rules:
1126.11/292.16	
1126.11/292.16	   rev1(0', nil) -> 0'
1126.11/292.16	   rev1(s, nil) -> s
1126.11/292.16	   rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	   rev(nil) -> nil
1126.11/292.16	   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	   rev2(X, nil) -> nil
1126.11/292.16	   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	S is empty.
1126.11/292.16	Rewrite Strategy: INNERMOST
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1126.11/292.16	Infered types.
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(6)
1126.11/292.16	Obligation:
1126.11/292.16	Innermost TRS:
1126.11/292.16	Rules:
1126.11/292.16	rev1(0', nil) -> 0'
1126.11/292.16	rev1(s, nil) -> s
1126.11/292.16	rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	rev(nil) -> nil
1126.11/292.16	rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	rev2(X, nil) -> nil
1126.11/292.16	rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	Types:
1126.11/292.16	rev1 :: 0':s -> nil:cons -> 0':s
1126.11/292.16	0' :: 0':s
1126.11/292.16	nil :: nil:cons
1126.11/292.16	s :: 0':s
1126.11/292.16	cons :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	rev :: nil:cons -> nil:cons
1126.11/292.16	rev2 :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	hole_0':s1_0 :: 0':s
1126.11/292.16	hole_nil:cons2_0 :: nil:cons
1126.11/292.16	gen_nil:cons3_0 :: Nat -> nil:cons
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(7) OrderProof (LOWER BOUND(ID))
1126.11/292.16	Heuristically decided to analyse the following defined symbols:
1126.11/292.16	rev1, rev, rev2
1126.11/292.16	
1126.11/292.16	They will be analysed ascendingly in the following order:
1126.11/292.16	rev1 < rev
1126.11/292.16	rev = rev2
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(8)
1126.11/292.16	Obligation:
1126.11/292.16	Innermost TRS:
1126.11/292.16	Rules:
1126.11/292.16	rev1(0', nil) -> 0'
1126.11/292.16	rev1(s, nil) -> s
1126.11/292.16	rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	rev(nil) -> nil
1126.11/292.16	rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	rev2(X, nil) -> nil
1126.11/292.16	rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	Types:
1126.11/292.16	rev1 :: 0':s -> nil:cons -> 0':s
1126.11/292.16	0' :: 0':s
1126.11/292.16	nil :: nil:cons
1126.11/292.16	s :: 0':s
1126.11/292.16	cons :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	rev :: nil:cons -> nil:cons
1126.11/292.16	rev2 :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	hole_0':s1_0 :: 0':s
1126.11/292.16	hole_nil:cons2_0 :: nil:cons
1126.11/292.16	gen_nil:cons3_0 :: Nat -> nil:cons
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	Generator Equations:
1126.11/292.16	gen_nil:cons3_0(0) <=> nil
1126.11/292.16	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	The following defined symbols remain to be analysed:
1126.11/292.16	rev1, rev, rev2
1126.11/292.16	
1126.11/292.16	They will be analysed ascendingly in the following order:
1126.11/292.16	rev1 < rev
1126.11/292.16	rev = rev2
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(9) RewriteLemmaProof (LOWER BOUND(ID))
1126.11/292.16	Proved the following rewrite lemma:
1126.11/292.16	rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)
1126.11/292.16	
1126.11/292.16	Induction Base:
1126.11/292.16	rev1(0', gen_nil:cons3_0(0)) ->_R^Omega(1)
1126.11/292.16	0'
1126.11/292.16	
1126.11/292.16	Induction Step:
1126.11/292.16	rev1(0', gen_nil:cons3_0(+(n5_0, 1))) ->_R^Omega(1)
1126.11/292.16	rev1(0', gen_nil:cons3_0(n5_0)) ->_IH
1126.11/292.16	0'
1126.11/292.16	
1126.11/292.16	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(10)
1126.11/292.16	Complex Obligation (BEST)
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(11)
1126.11/292.16	Obligation:
1126.11/292.16	Proved the lower bound n^1 for the following obligation:
1126.11/292.16	
1126.11/292.16	Innermost TRS:
1126.11/292.16	Rules:
1126.11/292.16	rev1(0', nil) -> 0'
1126.11/292.16	rev1(s, nil) -> s
1126.11/292.16	rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	rev(nil) -> nil
1126.11/292.16	rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	rev2(X, nil) -> nil
1126.11/292.16	rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	Types:
1126.11/292.16	rev1 :: 0':s -> nil:cons -> 0':s
1126.11/292.16	0' :: 0':s
1126.11/292.16	nil :: nil:cons
1126.11/292.16	s :: 0':s
1126.11/292.16	cons :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	rev :: nil:cons -> nil:cons
1126.11/292.16	rev2 :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	hole_0':s1_0 :: 0':s
1126.11/292.16	hole_nil:cons2_0 :: nil:cons
1126.11/292.16	gen_nil:cons3_0 :: Nat -> nil:cons
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	Generator Equations:
1126.11/292.16	gen_nil:cons3_0(0) <=> nil
1126.11/292.16	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	The following defined symbols remain to be analysed:
1126.11/292.16	rev1, rev, rev2
1126.11/292.16	
1126.11/292.16	They will be analysed ascendingly in the following order:
1126.11/292.16	rev1 < rev
1126.11/292.16	rev = rev2
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(12) LowerBoundPropagationProof (FINISHED)
1126.11/292.16	Propagated lower bound.
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(13)
1126.11/292.16	BOUNDS(n^1, INF)
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(14)
1126.11/292.16	Obligation:
1126.11/292.16	Innermost TRS:
1126.11/292.16	Rules:
1126.11/292.16	rev1(0', nil) -> 0'
1126.11/292.16	rev1(s, nil) -> s
1126.11/292.16	rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	rev(nil) -> nil
1126.11/292.16	rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	rev2(X, nil) -> nil
1126.11/292.16	rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	Types:
1126.11/292.16	rev1 :: 0':s -> nil:cons -> 0':s
1126.11/292.16	0' :: 0':s
1126.11/292.16	nil :: nil:cons
1126.11/292.16	s :: 0':s
1126.11/292.16	cons :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	rev :: nil:cons -> nil:cons
1126.11/292.16	rev2 :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	hole_0':s1_0 :: 0':s
1126.11/292.16	hole_nil:cons2_0 :: nil:cons
1126.11/292.16	gen_nil:cons3_0 :: Nat -> nil:cons
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	Lemmas:
1126.11/292.16	rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	Generator Equations:
1126.11/292.16	gen_nil:cons3_0(0) <=> nil
1126.11/292.16	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	The following defined symbols remain to be analysed:
1126.11/292.16	rev2, rev
1126.11/292.16	
1126.11/292.16	They will be analysed ascendingly in the following order:
1126.11/292.16	rev = rev2
1126.11/292.16	
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(15) RewriteLemmaProof (LOWER BOUND(ID))
1126.11/292.16	Proved the following rewrite lemma:
1126.11/292.16	rev2(0', gen_nil:cons3_0(+(1, n203_0))) -> *4_0, rt in Omega(n203_0)
1126.11/292.16	
1126.11/292.16	Induction Base:
1126.11/292.16	rev2(0', gen_nil:cons3_0(+(1, 0)))
1126.11/292.16	
1126.11/292.16	Induction Step:
1126.11/292.16	rev2(0', gen_nil:cons3_0(+(1, +(n203_0, 1)))) ->_R^Omega(1)
1126.11/292.16	rev(cons(0', rev(rev2(0', gen_nil:cons3_0(+(1, n203_0)))))) ->_IH
1126.11/292.16	rev(cons(0', rev(*4_0)))
1126.11/292.16	
1126.11/292.16	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.11/292.16	----------------------------------------
1126.11/292.16	
1126.11/292.16	(16)
1126.11/292.16	Obligation:
1126.11/292.16	Innermost TRS:
1126.11/292.16	Rules:
1126.11/292.16	rev1(0', nil) -> 0'
1126.11/292.16	rev1(s, nil) -> s
1126.11/292.16	rev1(X, cons(Y, L)) -> rev1(Y, L)
1126.11/292.16	rev(nil) -> nil
1126.11/292.16	rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
1126.11/292.16	rev2(X, nil) -> nil
1126.11/292.16	rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))
1126.11/292.16	
1126.11/292.16	Types:
1126.11/292.16	rev1 :: 0':s -> nil:cons -> 0':s
1126.11/292.16	0' :: 0':s
1126.11/292.16	nil :: nil:cons
1126.11/292.16	s :: 0':s
1126.11/292.16	cons :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	rev :: nil:cons -> nil:cons
1126.11/292.16	rev2 :: 0':s -> nil:cons -> nil:cons
1126.11/292.16	hole_0':s1_0 :: 0':s
1126.11/292.16	hole_nil:cons2_0 :: nil:cons
1126.11/292.16	gen_nil:cons3_0 :: Nat -> nil:cons
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	Lemmas:
1126.11/292.16	rev1(0', gen_nil:cons3_0(n5_0)) -> 0', rt in Omega(1 + n5_0)
1126.11/292.16	rev2(0', gen_nil:cons3_0(+(1, n203_0))) -> *4_0, rt in Omega(n203_0)
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	Generator Equations:
1126.11/292.16	gen_nil:cons3_0(0) <=> nil
1126.11/292.16	gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
1126.11/292.16	
1126.11/292.16	
1126.11/292.16	The following defined symbols remain to be analysed:
1126.11/292.16	rev
1126.11/292.16	
1126.11/292.16	They will be analysed ascendingly in the following order:
1126.11/292.16	rev = rev2
1126.80/292.35	EOF