1071.81/291.57	WORST_CASE(Omega(n^2), ?)
1072.04/291.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1072.04/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1072.04/291.58	
1072.04/291.58	(0) CpxTRS
1072.04/291.58	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1072.04/291.58	(2) CpxTRS
1072.04/291.58	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1072.04/291.58	(4) typed CpxTrs
1072.04/291.58	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1072.04/291.58	(6) typed CpxTrs
1072.04/291.58	(7) RewriteLemmaProof [LOWER BOUND(ID), 247 ms]
1072.04/291.58	(8) BEST
1072.04/291.58	    (9) proven lower bound
1072.04/291.58	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1072.04/291.58	        (11) BOUNDS(n^1, INF)
1072.04/291.58	    (12) typed CpxTrs
1072.04/291.58	        (13) RewriteLemmaProof [LOWER BOUND(ID), 67 ms]
1072.04/291.58	        (14) proven lower bound
1072.04/291.58	        (15) LowerBoundPropagationProof [FINISHED, 0 ms]
1072.04/291.58	        (16) BOUNDS(n^2, INF)
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(0)
1072.04/291.58	Obligation:
1072.04/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The TRS R consists of the following rules:
1072.04/291.58	
1072.04/291.58	   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	   sum(cons(0, x), y) -> sum(x, y)
1072.04/291.58	   sum(nil, y) -> y
1072.04/291.58	   weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0, x)))
1072.04/291.58	   weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	S is empty.
1072.04/291.58	Rewrite Strategy: INNERMOST
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1072.04/291.58	Renamed function symbols to avoid clashes with predefined symbol.
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(2)
1072.04/291.58	Obligation:
1072.04/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The TRS R consists of the following rules:
1072.04/291.58	
1072.04/291.58	   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	   sum(cons(0', x), y) -> sum(x, y)
1072.04/291.58	   sum(nil, y) -> y
1072.04/291.58	   weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1072.04/291.58	   weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	S is empty.
1072.04/291.58	Rewrite Strategy: INNERMOST
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1072.04/291.58	Infered types.
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(4)
1072.04/291.58	Obligation:
1072.04/291.58	Innermost TRS:
1072.04/291.58	Rules:
1072.04/291.58	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	sum(cons(0', x), y) -> sum(x, y)
1072.04/291.58	sum(nil, y) -> y
1072.04/291.58	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1072.04/291.58	weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	Types:
1072.04/291.58	sum :: cons:nil -> cons:nil -> cons:nil
1072.04/291.58	cons :: s:0' -> cons:nil -> cons:nil
1072.04/291.58	s :: s:0' -> s:0'
1072.04/291.58	0' :: s:0'
1072.04/291.58	nil :: cons:nil
1072.04/291.58	weight :: cons:nil -> s:0'
1072.04/291.58	hole_cons:nil1_0 :: cons:nil
1072.04/291.58	hole_s:0'2_0 :: s:0'
1072.04/291.58	gen_cons:nil3_0 :: Nat -> cons:nil
1072.04/291.58	gen_s:0'4_0 :: Nat -> s:0'
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(5) OrderProof (LOWER BOUND(ID))
1072.04/291.58	Heuristically decided to analyse the following defined symbols:
1072.04/291.58	sum, weight
1072.04/291.58	
1072.04/291.58	They will be analysed ascendingly in the following order:
1072.04/291.58	sum < weight
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(6)
1072.04/291.58	Obligation:
1072.04/291.58	Innermost TRS:
1072.04/291.58	Rules:
1072.04/291.58	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	sum(cons(0', x), y) -> sum(x, y)
1072.04/291.58	sum(nil, y) -> y
1072.04/291.58	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1072.04/291.58	weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	Types:
1072.04/291.58	sum :: cons:nil -> cons:nil -> cons:nil
1072.04/291.58	cons :: s:0' -> cons:nil -> cons:nil
1072.04/291.58	s :: s:0' -> s:0'
1072.04/291.58	0' :: s:0'
1072.04/291.58	nil :: cons:nil
1072.04/291.58	weight :: cons:nil -> s:0'
1072.04/291.58	hole_cons:nil1_0 :: cons:nil
1072.04/291.58	hole_s:0'2_0 :: s:0'
1072.04/291.58	gen_cons:nil3_0 :: Nat -> cons:nil
1072.04/291.58	gen_s:0'4_0 :: Nat -> s:0'
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	Generator Equations:
1072.04/291.58	gen_cons:nil3_0(0) <=> nil
1072.04/291.58	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1072.04/291.58	gen_s:0'4_0(0) <=> 0'
1072.04/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The following defined symbols remain to be analysed:
1072.04/291.58	sum, weight
1072.04/291.58	
1072.04/291.58	They will be analysed ascendingly in the following order:
1072.04/291.58	sum < weight
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(7) RewriteLemmaProof (LOWER BOUND(ID))
1072.04/291.58	Proved the following rewrite lemma:
1072.04/291.58	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) -> gen_cons:nil3_0(b), rt in Omega(1 + n6_0)
1072.04/291.58	
1072.04/291.58	Induction Base:
1072.04/291.58	sum(gen_cons:nil3_0(0), gen_cons:nil3_0(b)) ->_R^Omega(1)
1072.04/291.58	gen_cons:nil3_0(b)
1072.04/291.58	
1072.04/291.58	Induction Step:
1072.04/291.58	sum(gen_cons:nil3_0(+(n6_0, 1)), gen_cons:nil3_0(b)) ->_R^Omega(1)
1072.04/291.58	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) ->_IH
1072.04/291.58	gen_cons:nil3_0(b)
1072.04/291.58	
1072.04/291.58	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(8)
1072.04/291.58	Complex Obligation (BEST)
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(9)
1072.04/291.58	Obligation:
1072.04/291.58	Proved the lower bound n^1 for the following obligation:
1072.04/291.58	
1072.04/291.58	Innermost TRS:
1072.04/291.58	Rules:
1072.04/291.58	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	sum(cons(0', x), y) -> sum(x, y)
1072.04/291.58	sum(nil, y) -> y
1072.04/291.58	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1072.04/291.58	weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	Types:
1072.04/291.58	sum :: cons:nil -> cons:nil -> cons:nil
1072.04/291.58	cons :: s:0' -> cons:nil -> cons:nil
1072.04/291.58	s :: s:0' -> s:0'
1072.04/291.58	0' :: s:0'
1072.04/291.58	nil :: cons:nil
1072.04/291.58	weight :: cons:nil -> s:0'
1072.04/291.58	hole_cons:nil1_0 :: cons:nil
1072.04/291.58	hole_s:0'2_0 :: s:0'
1072.04/291.58	gen_cons:nil3_0 :: Nat -> cons:nil
1072.04/291.58	gen_s:0'4_0 :: Nat -> s:0'
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	Generator Equations:
1072.04/291.58	gen_cons:nil3_0(0) <=> nil
1072.04/291.58	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1072.04/291.58	gen_s:0'4_0(0) <=> 0'
1072.04/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The following defined symbols remain to be analysed:
1072.04/291.58	sum, weight
1072.04/291.58	
1072.04/291.58	They will be analysed ascendingly in the following order:
1072.04/291.58	sum < weight
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(10) LowerBoundPropagationProof (FINISHED)
1072.04/291.58	Propagated lower bound.
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(11)
1072.04/291.58	BOUNDS(n^1, INF)
1072.04/291.58	
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(12)
1072.04/291.58	Obligation:
1072.04/291.58	Innermost TRS:
1072.04/291.58	Rules:
1072.04/291.58	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	sum(cons(0', x), y) -> sum(x, y)
1072.04/291.58	sum(nil, y) -> y
1072.04/291.58	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1072.04/291.58	weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	Types:
1072.04/291.58	sum :: cons:nil -> cons:nil -> cons:nil
1072.04/291.58	cons :: s:0' -> cons:nil -> cons:nil
1072.04/291.58	s :: s:0' -> s:0'
1072.04/291.58	0' :: s:0'
1072.04/291.58	nil :: cons:nil
1072.04/291.58	weight :: cons:nil -> s:0'
1072.04/291.58	hole_cons:nil1_0 :: cons:nil
1072.04/291.58	hole_s:0'2_0 :: s:0'
1072.04/291.58	gen_cons:nil3_0 :: Nat -> cons:nil
1072.04/291.58	gen_s:0'4_0 :: Nat -> s:0'
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	Lemmas:
1072.04/291.58	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) -> gen_cons:nil3_0(b), rt in Omega(1 + n6_0)
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	Generator Equations:
1072.04/291.58	gen_cons:nil3_0(0) <=> nil
1072.04/291.58	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1072.04/291.58	gen_s:0'4_0(0) <=> 0'
1072.04/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The following defined symbols remain to be analysed:
1072.04/291.58	weight
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(13) RewriteLemmaProof (LOWER BOUND(ID))
1072.04/291.58	Proved the following rewrite lemma:
1072.04/291.58	weight(gen_cons:nil3_0(+(1, n535_0))) -> gen_s:0'4_0(0), rt in Omega(1 + n535_0 + n535_0^2)
1072.04/291.58	
1072.04/291.58	Induction Base:
1072.04/291.58	weight(gen_cons:nil3_0(+(1, 0))) ->_R^Omega(1)
1072.04/291.58	0'
1072.04/291.58	
1072.04/291.58	Induction Step:
1072.04/291.58	weight(gen_cons:nil3_0(+(1, +(n535_0, 1)))) ->_R^Omega(1)
1072.04/291.58	weight(sum(cons(0', cons(0', gen_cons:nil3_0(n535_0))), cons(0', gen_cons:nil3_0(n535_0)))) ->_L^Omega(3 + n535_0)
1072.04/291.58	weight(gen_cons:nil3_0(+(n535_0, 1))) ->_IH
1072.04/291.58	gen_s:0'4_0(0)
1072.04/291.58	
1072.04/291.58	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(14)
1072.04/291.58	Obligation:
1072.04/291.58	Proved the lower bound n^2 for the following obligation:
1072.04/291.58	
1072.04/291.58	Innermost TRS:
1072.04/291.58	Rules:
1072.04/291.58	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1072.04/291.58	sum(cons(0', x), y) -> sum(x, y)
1072.04/291.58	sum(nil, y) -> y
1072.04/291.58	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1072.04/291.58	weight(cons(n, nil)) -> n
1072.04/291.58	
1072.04/291.58	Types:
1072.04/291.58	sum :: cons:nil -> cons:nil -> cons:nil
1072.04/291.58	cons :: s:0' -> cons:nil -> cons:nil
1072.04/291.58	s :: s:0' -> s:0'
1072.04/291.58	0' :: s:0'
1072.04/291.58	nil :: cons:nil
1072.04/291.58	weight :: cons:nil -> s:0'
1072.04/291.58	hole_cons:nil1_0 :: cons:nil
1072.04/291.58	hole_s:0'2_0 :: s:0'
1072.04/291.58	gen_cons:nil3_0 :: Nat -> cons:nil
1072.04/291.58	gen_s:0'4_0 :: Nat -> s:0'
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	Lemmas:
1072.04/291.58	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) -> gen_cons:nil3_0(b), rt in Omega(1 + n6_0)
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	Generator Equations:
1072.04/291.58	gen_cons:nil3_0(0) <=> nil
1072.04/291.58	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1072.04/291.58	gen_s:0'4_0(0) <=> 0'
1072.04/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1072.04/291.58	
1072.04/291.58	
1072.04/291.58	The following defined symbols remain to be analysed:
1072.04/291.58	weight
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(15) LowerBoundPropagationProof (FINISHED)
1072.04/291.58	Propagated lower bound.
1072.04/291.58	----------------------------------------
1072.04/291.58	
1072.04/291.58	(16)
1072.04/291.58	BOUNDS(n^2, INF)
1072.04/291.64	EOF