1071.75/291.52	WORST_CASE(Omega(n^2), ?)
1071.90/291.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1071.90/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1071.90/291.57	
1071.90/291.57	(0) CpxTRS
1071.90/291.57	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1071.90/291.57	(2) CpxTRS
1071.90/291.57	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1071.90/291.57	(4) typed CpxTrs
1071.90/291.57	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1071.90/291.57	(6) typed CpxTrs
1071.90/291.57	(7) RewriteLemmaProof [LOWER BOUND(ID), 243 ms]
1071.90/291.57	(8) BEST
1071.90/291.57	    (9) proven lower bound
1071.90/291.57	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1071.90/291.57	        (11) BOUNDS(n^1, INF)
1071.90/291.57	    (12) typed CpxTrs
1071.90/291.57	        (13) RewriteLemmaProof [LOWER BOUND(ID), 36 ms]
1071.90/291.57	        (14) proven lower bound
1071.90/291.57	        (15) LowerBoundPropagationProof [FINISHED, 0 ms]
1071.90/291.57	        (16) BOUNDS(n^2, INF)
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(0)
1071.90/291.57	Obligation:
1071.90/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The TRS R consists of the following rules:
1071.90/291.57	
1071.90/291.57	   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	   sum(cons(0, x), y) -> sum(x, y)
1071.90/291.57	   sum(nil, y) -> y
1071.90/291.57	   weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0, x)))
1071.90/291.57	   weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	S is empty.
1071.90/291.57	Rewrite Strategy: FULL
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1071.90/291.57	Renamed function symbols to avoid clashes with predefined symbol.
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(2)
1071.90/291.57	Obligation:
1071.90/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The TRS R consists of the following rules:
1071.90/291.57	
1071.90/291.57	   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	   sum(cons(0', x), y) -> sum(x, y)
1071.90/291.57	   sum(nil, y) -> y
1071.90/291.57	   weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1071.90/291.57	   weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	S is empty.
1071.90/291.57	Rewrite Strategy: FULL
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1071.90/291.57	Infered types.
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(4)
1071.90/291.57	Obligation:
1071.90/291.57	TRS:
1071.90/291.57	Rules:
1071.90/291.57	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	sum(cons(0', x), y) -> sum(x, y)
1071.90/291.57	sum(nil, y) -> y
1071.90/291.57	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1071.90/291.57	weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	Types:
1071.90/291.57	sum :: cons:nil -> cons:nil -> cons:nil
1071.90/291.57	cons :: s:0' -> cons:nil -> cons:nil
1071.90/291.57	s :: s:0' -> s:0'
1071.90/291.57	0' :: s:0'
1071.90/291.57	nil :: cons:nil
1071.90/291.57	weight :: cons:nil -> s:0'
1071.90/291.57	hole_cons:nil1_0 :: cons:nil
1071.90/291.57	hole_s:0'2_0 :: s:0'
1071.90/291.57	gen_cons:nil3_0 :: Nat -> cons:nil
1071.90/291.57	gen_s:0'4_0 :: Nat -> s:0'
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(5) OrderProof (LOWER BOUND(ID))
1071.90/291.57	Heuristically decided to analyse the following defined symbols:
1071.90/291.57	sum, weight
1071.90/291.57	
1071.90/291.57	They will be analysed ascendingly in the following order:
1071.90/291.57	sum < weight
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(6)
1071.90/291.57	Obligation:
1071.90/291.57	TRS:
1071.90/291.57	Rules:
1071.90/291.57	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	sum(cons(0', x), y) -> sum(x, y)
1071.90/291.57	sum(nil, y) -> y
1071.90/291.57	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1071.90/291.57	weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	Types:
1071.90/291.57	sum :: cons:nil -> cons:nil -> cons:nil
1071.90/291.57	cons :: s:0' -> cons:nil -> cons:nil
1071.90/291.57	s :: s:0' -> s:0'
1071.90/291.57	0' :: s:0'
1071.90/291.57	nil :: cons:nil
1071.90/291.57	weight :: cons:nil -> s:0'
1071.90/291.57	hole_cons:nil1_0 :: cons:nil
1071.90/291.57	hole_s:0'2_0 :: s:0'
1071.90/291.57	gen_cons:nil3_0 :: Nat -> cons:nil
1071.90/291.57	gen_s:0'4_0 :: Nat -> s:0'
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	Generator Equations:
1071.90/291.57	gen_cons:nil3_0(0) <=> nil
1071.90/291.57	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1071.90/291.57	gen_s:0'4_0(0) <=> 0'
1071.90/291.57	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The following defined symbols remain to be analysed:
1071.90/291.57	sum, weight
1071.90/291.57	
1071.90/291.57	They will be analysed ascendingly in the following order:
1071.90/291.57	sum < weight
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(7) RewriteLemmaProof (LOWER BOUND(ID))
1071.90/291.57	Proved the following rewrite lemma:
1071.90/291.57	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) -> gen_cons:nil3_0(b), rt in Omega(1 + n6_0)
1071.90/291.57	
1071.90/291.57	Induction Base:
1071.90/291.57	sum(gen_cons:nil3_0(0), gen_cons:nil3_0(b)) ->_R^Omega(1)
1071.90/291.57	gen_cons:nil3_0(b)
1071.90/291.57	
1071.90/291.57	Induction Step:
1071.90/291.57	sum(gen_cons:nil3_0(+(n6_0, 1)), gen_cons:nil3_0(b)) ->_R^Omega(1)
1071.90/291.57	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) ->_IH
1071.90/291.57	gen_cons:nil3_0(b)
1071.90/291.57	
1071.90/291.57	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(8)
1071.90/291.57	Complex Obligation (BEST)
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(9)
1071.90/291.57	Obligation:
1071.90/291.57	Proved the lower bound n^1 for the following obligation:
1071.90/291.57	
1071.90/291.57	TRS:
1071.90/291.57	Rules:
1071.90/291.57	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	sum(cons(0', x), y) -> sum(x, y)
1071.90/291.57	sum(nil, y) -> y
1071.90/291.57	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1071.90/291.57	weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	Types:
1071.90/291.57	sum :: cons:nil -> cons:nil -> cons:nil
1071.90/291.57	cons :: s:0' -> cons:nil -> cons:nil
1071.90/291.57	s :: s:0' -> s:0'
1071.90/291.57	0' :: s:0'
1071.90/291.57	nil :: cons:nil
1071.90/291.57	weight :: cons:nil -> s:0'
1071.90/291.57	hole_cons:nil1_0 :: cons:nil
1071.90/291.57	hole_s:0'2_0 :: s:0'
1071.90/291.57	gen_cons:nil3_0 :: Nat -> cons:nil
1071.90/291.57	gen_s:0'4_0 :: Nat -> s:0'
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	Generator Equations:
1071.90/291.57	gen_cons:nil3_0(0) <=> nil
1071.90/291.57	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1071.90/291.57	gen_s:0'4_0(0) <=> 0'
1071.90/291.57	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The following defined symbols remain to be analysed:
1071.90/291.57	sum, weight
1071.90/291.57	
1071.90/291.57	They will be analysed ascendingly in the following order:
1071.90/291.57	sum < weight
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(10) LowerBoundPropagationProof (FINISHED)
1071.90/291.57	Propagated lower bound.
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(11)
1071.90/291.57	BOUNDS(n^1, INF)
1071.90/291.57	
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(12)
1071.90/291.57	Obligation:
1071.90/291.57	TRS:
1071.90/291.57	Rules:
1071.90/291.57	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	sum(cons(0', x), y) -> sum(x, y)
1071.90/291.57	sum(nil, y) -> y
1071.90/291.57	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1071.90/291.57	weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	Types:
1071.90/291.57	sum :: cons:nil -> cons:nil -> cons:nil
1071.90/291.57	cons :: s:0' -> cons:nil -> cons:nil
1071.90/291.57	s :: s:0' -> s:0'
1071.90/291.57	0' :: s:0'
1071.90/291.57	nil :: cons:nil
1071.90/291.57	weight :: cons:nil -> s:0'
1071.90/291.57	hole_cons:nil1_0 :: cons:nil
1071.90/291.57	hole_s:0'2_0 :: s:0'
1071.90/291.57	gen_cons:nil3_0 :: Nat -> cons:nil
1071.90/291.57	gen_s:0'4_0 :: Nat -> s:0'
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	Lemmas:
1071.90/291.57	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) -> gen_cons:nil3_0(b), rt in Omega(1 + n6_0)
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	Generator Equations:
1071.90/291.57	gen_cons:nil3_0(0) <=> nil
1071.90/291.57	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1071.90/291.57	gen_s:0'4_0(0) <=> 0'
1071.90/291.57	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The following defined symbols remain to be analysed:
1071.90/291.57	weight
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(13) RewriteLemmaProof (LOWER BOUND(ID))
1071.90/291.57	Proved the following rewrite lemma:
1071.90/291.57	weight(gen_cons:nil3_0(+(1, n511_0))) -> gen_s:0'4_0(0), rt in Omega(1 + n511_0 + n511_0^2)
1071.90/291.57	
1071.90/291.57	Induction Base:
1071.90/291.57	weight(gen_cons:nil3_0(+(1, 0))) ->_R^Omega(1)
1071.90/291.57	0'
1071.90/291.57	
1071.90/291.57	Induction Step:
1071.90/291.57	weight(gen_cons:nil3_0(+(1, +(n511_0, 1)))) ->_R^Omega(1)
1071.90/291.57	weight(sum(cons(0', cons(0', gen_cons:nil3_0(n511_0))), cons(0', gen_cons:nil3_0(n511_0)))) ->_L^Omega(3 + n511_0)
1071.90/291.57	weight(gen_cons:nil3_0(+(n511_0, 1))) ->_IH
1071.90/291.57	gen_s:0'4_0(0)
1071.90/291.57	
1071.90/291.57	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(14)
1071.90/291.57	Obligation:
1071.90/291.57	Proved the lower bound n^2 for the following obligation:
1071.90/291.57	
1071.90/291.57	TRS:
1071.90/291.57	Rules:
1071.90/291.57	sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
1071.90/291.57	sum(cons(0', x), y) -> sum(x, y)
1071.90/291.57	sum(nil, y) -> y
1071.90/291.57	weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0', x)))
1071.90/291.57	weight(cons(n, nil)) -> n
1071.90/291.57	
1071.90/291.57	Types:
1071.90/291.57	sum :: cons:nil -> cons:nil -> cons:nil
1071.90/291.57	cons :: s:0' -> cons:nil -> cons:nil
1071.90/291.57	s :: s:0' -> s:0'
1071.90/291.57	0' :: s:0'
1071.90/291.57	nil :: cons:nil
1071.90/291.57	weight :: cons:nil -> s:0'
1071.90/291.57	hole_cons:nil1_0 :: cons:nil
1071.90/291.57	hole_s:0'2_0 :: s:0'
1071.90/291.57	gen_cons:nil3_0 :: Nat -> cons:nil
1071.90/291.57	gen_s:0'4_0 :: Nat -> s:0'
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	Lemmas:
1071.90/291.57	sum(gen_cons:nil3_0(n6_0), gen_cons:nil3_0(b)) -> gen_cons:nil3_0(b), rt in Omega(1 + n6_0)
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	Generator Equations:
1071.90/291.57	gen_cons:nil3_0(0) <=> nil
1071.90/291.57	gen_cons:nil3_0(+(x, 1)) <=> cons(0', gen_cons:nil3_0(x))
1071.90/291.57	gen_s:0'4_0(0) <=> 0'
1071.90/291.57	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
1071.90/291.57	
1071.90/291.57	
1071.90/291.57	The following defined symbols remain to be analysed:
1071.90/291.57	weight
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(15) LowerBoundPropagationProof (FINISHED)
1071.90/291.57	Propagated lower bound.
1071.90/291.57	----------------------------------------
1071.90/291.57	
1071.90/291.57	(16)
1071.90/291.57	BOUNDS(n^2, INF)
1072.09/291.64	EOF