1127.08/291.50	WORST_CASE(Omega(n^1), ?)
1127.23/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1127.23/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1127.23/291.56	
1127.23/291.56	(0) CpxTRS
1127.23/291.56	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1127.23/291.56	(2) CpxTRS
1127.23/291.56	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1127.23/291.56	(4) typed CpxTrs
1127.23/291.56	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1127.23/291.56	(6) typed CpxTrs
1127.23/291.56	(7) RewriteLemmaProof [LOWER BOUND(ID), 219 ms]
1127.23/291.56	(8) BEST
1127.23/291.56	    (9) proven lower bound
1127.23/291.56	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1127.23/291.56	        (11) BOUNDS(n^1, INF)
1127.23/291.56	    (12) typed CpxTrs
1127.23/291.56	        (13) RewriteLemmaProof [LOWER BOUND(ID), 556 ms]
1127.23/291.56	        (14) typed CpxTrs
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(0)
1127.23/291.56	Obligation:
1127.23/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The TRS R consists of the following rules:
1127.23/291.56	
1127.23/291.56	   rev(nil) -> nil
1127.23/291.56	   rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	   rev1(x, nil) -> x
1127.23/291.56	   rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	   rev2(x, nil) -> nil
1127.23/291.56	   rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	S is empty.
1127.23/291.56	Rewrite Strategy: INNERMOST
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1127.23/291.56	Renamed function symbols to avoid clashes with predefined symbol.
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(2)
1127.23/291.56	Obligation:
1127.23/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The TRS R consists of the following rules:
1127.23/291.56	
1127.23/291.56	   rev(nil) -> nil
1127.23/291.56	   rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	   rev1(x, nil) -> x
1127.23/291.56	   rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	   rev2(x, nil) -> nil
1127.23/291.56	   rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	S is empty.
1127.23/291.56	Rewrite Strategy: INNERMOST
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1127.23/291.56	Infered types.
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(4)
1127.23/291.56	Obligation:
1127.23/291.56	Innermost TRS:
1127.23/291.56	Rules:
1127.23/291.56	rev(nil) -> nil
1127.23/291.56	rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	rev1(x, nil) -> x
1127.23/291.56	rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	rev2(x, nil) -> nil
1127.23/291.56	rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	Types:
1127.23/291.56	rev :: nil:++ -> nil:++
1127.23/291.56	nil :: nil:++
1127.23/291.56	++ :: rev1 -> nil:++ -> nil:++
1127.23/291.56	rev1 :: rev1 -> nil:++ -> rev1
1127.23/291.56	rev2 :: rev1 -> nil:++ -> nil:++
1127.23/291.56	hole_nil:++1_0 :: nil:++
1127.23/291.56	hole_rev12_0 :: rev1
1127.23/291.56	gen_nil:++3_0 :: Nat -> nil:++
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(5) OrderProof (LOWER BOUND(ID))
1127.23/291.56	Heuristically decided to analyse the following defined symbols:
1127.23/291.56	rev, rev1, rev2
1127.23/291.56	
1127.23/291.56	They will be analysed ascendingly in the following order:
1127.23/291.56	rev1 < rev
1127.23/291.56	rev = rev2
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(6)
1127.23/291.56	Obligation:
1127.23/291.56	Innermost TRS:
1127.23/291.56	Rules:
1127.23/291.56	rev(nil) -> nil
1127.23/291.56	rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	rev1(x, nil) -> x
1127.23/291.56	rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	rev2(x, nil) -> nil
1127.23/291.56	rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	Types:
1127.23/291.56	rev :: nil:++ -> nil:++
1127.23/291.56	nil :: nil:++
1127.23/291.56	++ :: rev1 -> nil:++ -> nil:++
1127.23/291.56	rev1 :: rev1 -> nil:++ -> rev1
1127.23/291.56	rev2 :: rev1 -> nil:++ -> nil:++
1127.23/291.56	hole_nil:++1_0 :: nil:++
1127.23/291.56	hole_rev12_0 :: rev1
1127.23/291.56	gen_nil:++3_0 :: Nat -> nil:++
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	Generator Equations:
1127.23/291.56	gen_nil:++3_0(0) <=> nil
1127.23/291.56	gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The following defined symbols remain to be analysed:
1127.23/291.56	rev1, rev, rev2
1127.23/291.56	
1127.23/291.56	They will be analysed ascendingly in the following order:
1127.23/291.56	rev1 < rev
1127.23/291.56	rev = rev2
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(7) RewriteLemmaProof (LOWER BOUND(ID))
1127.23/291.56	Proved the following rewrite lemma:
1127.23/291.56	rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) -> hole_rev12_0, rt in Omega(1 + n5_0)
1127.23/291.56	
1127.23/291.56	Induction Base:
1127.23/291.56	rev1(hole_rev12_0, gen_nil:++3_0(0)) ->_R^Omega(1)
1127.23/291.56	hole_rev12_0
1127.23/291.56	
1127.23/291.56	Induction Step:
1127.23/291.56	rev1(hole_rev12_0, gen_nil:++3_0(+(n5_0, 1))) ->_R^Omega(1)
1127.23/291.56	rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) ->_IH
1127.23/291.56	hole_rev12_0
1127.23/291.56	
1127.23/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(8)
1127.23/291.56	Complex Obligation (BEST)
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(9)
1127.23/291.56	Obligation:
1127.23/291.56	Proved the lower bound n^1 for the following obligation:
1127.23/291.56	
1127.23/291.56	Innermost TRS:
1127.23/291.56	Rules:
1127.23/291.56	rev(nil) -> nil
1127.23/291.56	rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	rev1(x, nil) -> x
1127.23/291.56	rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	rev2(x, nil) -> nil
1127.23/291.56	rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	Types:
1127.23/291.56	rev :: nil:++ -> nil:++
1127.23/291.56	nil :: nil:++
1127.23/291.56	++ :: rev1 -> nil:++ -> nil:++
1127.23/291.56	rev1 :: rev1 -> nil:++ -> rev1
1127.23/291.56	rev2 :: rev1 -> nil:++ -> nil:++
1127.23/291.56	hole_nil:++1_0 :: nil:++
1127.23/291.56	hole_rev12_0 :: rev1
1127.23/291.56	gen_nil:++3_0 :: Nat -> nil:++
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	Generator Equations:
1127.23/291.56	gen_nil:++3_0(0) <=> nil
1127.23/291.56	gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The following defined symbols remain to be analysed:
1127.23/291.56	rev1, rev, rev2
1127.23/291.56	
1127.23/291.56	They will be analysed ascendingly in the following order:
1127.23/291.56	rev1 < rev
1127.23/291.56	rev = rev2
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(10) LowerBoundPropagationProof (FINISHED)
1127.23/291.56	Propagated lower bound.
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(11)
1127.23/291.56	BOUNDS(n^1, INF)
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(12)
1127.23/291.56	Obligation:
1127.23/291.56	Innermost TRS:
1127.23/291.56	Rules:
1127.23/291.56	rev(nil) -> nil
1127.23/291.56	rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	rev1(x, nil) -> x
1127.23/291.56	rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	rev2(x, nil) -> nil
1127.23/291.56	rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	Types:
1127.23/291.56	rev :: nil:++ -> nil:++
1127.23/291.56	nil :: nil:++
1127.23/291.56	++ :: rev1 -> nil:++ -> nil:++
1127.23/291.56	rev1 :: rev1 -> nil:++ -> rev1
1127.23/291.56	rev2 :: rev1 -> nil:++ -> nil:++
1127.23/291.56	hole_nil:++1_0 :: nil:++
1127.23/291.56	hole_rev12_0 :: rev1
1127.23/291.56	gen_nil:++3_0 :: Nat -> nil:++
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	Lemmas:
1127.23/291.56	rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) -> hole_rev12_0, rt in Omega(1 + n5_0)
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	Generator Equations:
1127.23/291.56	gen_nil:++3_0(0) <=> nil
1127.23/291.56	gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The following defined symbols remain to be analysed:
1127.23/291.56	rev2, rev
1127.23/291.56	
1127.23/291.56	They will be analysed ascendingly in the following order:
1127.23/291.56	rev = rev2
1127.23/291.56	
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(13) RewriteLemmaProof (LOWER BOUND(ID))
1127.23/291.56	Proved the following rewrite lemma:
1127.23/291.56	rev2(hole_rev12_0, gen_nil:++3_0(+(1, n128_0))) -> *4_0, rt in Omega(n128_0)
1127.23/291.56	
1127.23/291.56	Induction Base:
1127.23/291.56	rev2(hole_rev12_0, gen_nil:++3_0(+(1, 0)))
1127.23/291.56	
1127.23/291.56	Induction Step:
1127.23/291.56	rev2(hole_rev12_0, gen_nil:++3_0(+(1, +(n128_0, 1)))) ->_R^Omega(1)
1127.23/291.56	rev(++(hole_rev12_0, rev(rev2(hole_rev12_0, gen_nil:++3_0(+(1, n128_0)))))) ->_IH
1127.23/291.56	rev(++(hole_rev12_0, rev(*4_0)))
1127.23/291.56	
1127.23/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1127.23/291.56	----------------------------------------
1127.23/291.56	
1127.23/291.56	(14)
1127.23/291.56	Obligation:
1127.23/291.56	Innermost TRS:
1127.23/291.56	Rules:
1127.23/291.56	rev(nil) -> nil
1127.23/291.56	rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
1127.23/291.56	rev1(x, nil) -> x
1127.23/291.56	rev1(x, ++(y, z)) -> rev1(y, z)
1127.23/291.56	rev2(x, nil) -> nil
1127.23/291.56	rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
1127.23/291.56	
1127.23/291.56	Types:
1127.23/291.56	rev :: nil:++ -> nil:++
1127.23/291.56	nil :: nil:++
1127.23/291.56	++ :: rev1 -> nil:++ -> nil:++
1127.23/291.56	rev1 :: rev1 -> nil:++ -> rev1
1127.23/291.56	rev2 :: rev1 -> nil:++ -> nil:++
1127.23/291.56	hole_nil:++1_0 :: nil:++
1127.23/291.56	hole_rev12_0 :: rev1
1127.23/291.56	gen_nil:++3_0 :: Nat -> nil:++
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	Lemmas:
1127.23/291.56	rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) -> hole_rev12_0, rt in Omega(1 + n5_0)
1127.23/291.56	rev2(hole_rev12_0, gen_nil:++3_0(+(1, n128_0))) -> *4_0, rt in Omega(n128_0)
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	Generator Equations:
1127.23/291.56	gen_nil:++3_0(0) <=> nil
1127.23/291.56	gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))
1127.23/291.56	
1127.23/291.56	
1127.23/291.56	The following defined symbols remain to be analysed:
1127.23/291.56	rev
1127.23/291.56	
1127.23/291.56	They will be analysed ascendingly in the following order:
1127.23/291.56	rev = rev2
1127.83/291.76	EOF