1106.81/291.56	WORST_CASE(Omega(n^1), ?)
1106.81/291.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1106.81/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1106.81/291.58	
1106.81/291.58	(0) CpxTRS
1106.81/291.58	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1106.81/291.58	(2) CpxTRS
1106.81/291.58	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1106.81/291.58	(4) typed CpxTrs
1106.81/291.58	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1106.81/291.58	(6) typed CpxTrs
1106.81/291.58	(7) RewriteLemmaProof [LOWER BOUND(ID), 230 ms]
1106.81/291.58	(8) BEST
1106.81/291.58	    (9) proven lower bound
1106.81/291.58	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1106.81/291.58	        (11) BOUNDS(n^1, INF)
1106.81/291.58	    (12) typed CpxTrs
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(0)
1106.81/291.58	Obligation:
1106.81/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	The TRS R consists of the following rules:
1106.81/291.58	
1106.81/291.58	   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
1106.81/291.58	   a____(X, nil) -> mark(X)
1106.81/291.58	   a____(nil, X) -> mark(X)
1106.81/291.58	   a__and(tt, X) -> mark(X)
1106.81/291.58	   a__isNePal(__(I, __(P, I))) -> tt
1106.81/291.58	   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
1106.81/291.58	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
1106.81/291.58	   mark(isNePal(X)) -> a__isNePal(mark(X))
1106.81/291.58	   mark(nil) -> nil
1106.81/291.58	   mark(tt) -> tt
1106.81/291.58	   a____(X1, X2) -> __(X1, X2)
1106.81/291.58	   a__and(X1, X2) -> and(X1, X2)
1106.81/291.58	   a__isNePal(X) -> isNePal(X)
1106.81/291.58	
1106.81/291.58	S is empty.
1106.81/291.58	Rewrite Strategy: FULL
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1106.81/291.58	Renamed function symbols to avoid clashes with predefined symbol.
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(2)
1106.81/291.58	Obligation:
1106.81/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	The TRS R consists of the following rules:
1106.81/291.58	
1106.81/291.58	   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
1106.81/291.58	   a____(X, nil) -> mark(X)
1106.81/291.58	   a____(nil, X) -> mark(X)
1106.81/291.58	   a__and(tt, X) -> mark(X)
1106.81/291.58	   a__isNePal(__(I, __(P, I))) -> tt
1106.81/291.58	   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
1106.81/291.58	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
1106.81/291.58	   mark(isNePal(X)) -> a__isNePal(mark(X))
1106.81/291.58	   mark(nil) -> nil
1106.81/291.58	   mark(tt) -> tt
1106.81/291.58	   a____(X1, X2) -> __(X1, X2)
1106.81/291.58	   a__and(X1, X2) -> and(X1, X2)
1106.81/291.58	   a__isNePal(X) -> isNePal(X)
1106.81/291.58	
1106.81/291.58	S is empty.
1106.81/291.58	Rewrite Strategy: FULL
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1106.81/291.58	Infered types.
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(4)
1106.81/291.58	Obligation:
1106.81/291.58	TRS:
1106.81/291.58	Rules:
1106.81/291.58	a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
1106.81/291.58	a____(X, nil) -> mark(X)
1106.81/291.58	a____(nil, X) -> mark(X)
1106.81/291.58	a__and(tt, X) -> mark(X)
1106.81/291.58	a__isNePal(__(I, __(P, I))) -> tt
1106.81/291.58	mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
1106.81/291.58	mark(and(X1, X2)) -> a__and(mark(X1), X2)
1106.81/291.58	mark(isNePal(X)) -> a__isNePal(mark(X))
1106.81/291.58	mark(nil) -> nil
1106.81/291.58	mark(tt) -> tt
1106.81/291.58	a____(X1, X2) -> __(X1, X2)
1106.81/291.58	a__and(X1, X2) -> and(X1, X2)
1106.81/291.58	a__isNePal(X) -> isNePal(X)
1106.81/291.58	
1106.81/291.58	Types:
1106.81/291.58	a____ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	__ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	mark :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	nil :: __:nil:tt:and:isNePal
1106.81/291.58	a__and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	tt :: __:nil:tt:and:isNePal
1106.81/291.58	a__isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	hole___:nil:tt:and:isNePal1_0 :: __:nil:tt:and:isNePal
1106.81/291.58	gen___:nil:tt:and:isNePal2_0 :: Nat -> __:nil:tt:and:isNePal
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(5) OrderProof (LOWER BOUND(ID))
1106.81/291.58	Heuristically decided to analyse the following defined symbols:
1106.81/291.58	a____, mark
1106.81/291.58	
1106.81/291.58	They will be analysed ascendingly in the following order:
1106.81/291.58	a____ = mark
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(6)
1106.81/291.58	Obligation:
1106.81/291.58	TRS:
1106.81/291.58	Rules:
1106.81/291.58	a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
1106.81/291.58	a____(X, nil) -> mark(X)
1106.81/291.58	a____(nil, X) -> mark(X)
1106.81/291.58	a__and(tt, X) -> mark(X)
1106.81/291.58	a__isNePal(__(I, __(P, I))) -> tt
1106.81/291.58	mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
1106.81/291.58	mark(and(X1, X2)) -> a__and(mark(X1), X2)
1106.81/291.58	mark(isNePal(X)) -> a__isNePal(mark(X))
1106.81/291.58	mark(nil) -> nil
1106.81/291.58	mark(tt) -> tt
1106.81/291.58	a____(X1, X2) -> __(X1, X2)
1106.81/291.58	a__and(X1, X2) -> and(X1, X2)
1106.81/291.58	a__isNePal(X) -> isNePal(X)
1106.81/291.58	
1106.81/291.58	Types:
1106.81/291.58	a____ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	__ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	mark :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	nil :: __:nil:tt:and:isNePal
1106.81/291.58	a__and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	tt :: __:nil:tt:and:isNePal
1106.81/291.58	a__isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	hole___:nil:tt:and:isNePal1_0 :: __:nil:tt:and:isNePal
1106.81/291.58	gen___:nil:tt:and:isNePal2_0 :: Nat -> __:nil:tt:and:isNePal
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	Generator Equations:
1106.81/291.58	gen___:nil:tt:and:isNePal2_0(0) <=> nil
1106.81/291.58	gen___:nil:tt:and:isNePal2_0(+(x, 1)) <=> __(nil, gen___:nil:tt:and:isNePal2_0(x))
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	The following defined symbols remain to be analysed:
1106.81/291.58	mark, a____
1106.81/291.58	
1106.81/291.58	They will be analysed ascendingly in the following order:
1106.81/291.58	a____ = mark
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(7) RewriteLemmaProof (LOWER BOUND(ID))
1106.81/291.58	Proved the following rewrite lemma:
1106.81/291.58	mark(gen___:nil:tt:and:isNePal2_0(n4_0)) -> gen___:nil:tt:and:isNePal2_0(0), rt in Omega(1 + n4_0)
1106.81/291.58	
1106.81/291.58	Induction Base:
1106.81/291.58	mark(gen___:nil:tt:and:isNePal2_0(0)) ->_R^Omega(1)
1106.81/291.58	nil
1106.81/291.58	
1106.81/291.58	Induction Step:
1106.81/291.58	mark(gen___:nil:tt:and:isNePal2_0(+(n4_0, 1))) ->_R^Omega(1)
1106.81/291.58	a____(mark(nil), mark(gen___:nil:tt:and:isNePal2_0(n4_0))) ->_R^Omega(1)
1106.81/291.58	a____(nil, mark(gen___:nil:tt:and:isNePal2_0(n4_0))) ->_IH
1106.81/291.58	a____(nil, gen___:nil:tt:and:isNePal2_0(0)) ->_R^Omega(1)
1106.81/291.58	mark(nil) ->_R^Omega(1)
1106.81/291.58	nil
1106.81/291.58	
1106.81/291.58	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(8)
1106.81/291.58	Complex Obligation (BEST)
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(9)
1106.81/291.58	Obligation:
1106.81/291.58	Proved the lower bound n^1 for the following obligation:
1106.81/291.58	
1106.81/291.58	TRS:
1106.81/291.58	Rules:
1106.81/291.58	a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
1106.81/291.58	a____(X, nil) -> mark(X)
1106.81/291.58	a____(nil, X) -> mark(X)
1106.81/291.58	a__and(tt, X) -> mark(X)
1106.81/291.58	a__isNePal(__(I, __(P, I))) -> tt
1106.81/291.58	mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
1106.81/291.58	mark(and(X1, X2)) -> a__and(mark(X1), X2)
1106.81/291.58	mark(isNePal(X)) -> a__isNePal(mark(X))
1106.81/291.58	mark(nil) -> nil
1106.81/291.58	mark(tt) -> tt
1106.81/291.58	a____(X1, X2) -> __(X1, X2)
1106.81/291.58	a__and(X1, X2) -> and(X1, X2)
1106.81/291.58	a__isNePal(X) -> isNePal(X)
1106.81/291.58	
1106.81/291.58	Types:
1106.81/291.58	a____ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	__ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	mark :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	nil :: __:nil:tt:and:isNePal
1106.81/291.58	a__and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	tt :: __:nil:tt:and:isNePal
1106.81/291.58	a__isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	hole___:nil:tt:and:isNePal1_0 :: __:nil:tt:and:isNePal
1106.81/291.58	gen___:nil:tt:and:isNePal2_0 :: Nat -> __:nil:tt:and:isNePal
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	Generator Equations:
1106.81/291.58	gen___:nil:tt:and:isNePal2_0(0) <=> nil
1106.81/291.58	gen___:nil:tt:and:isNePal2_0(+(x, 1)) <=> __(nil, gen___:nil:tt:and:isNePal2_0(x))
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	The following defined symbols remain to be analysed:
1106.81/291.58	mark, a____
1106.81/291.58	
1106.81/291.58	They will be analysed ascendingly in the following order:
1106.81/291.58	a____ = mark
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(10) LowerBoundPropagationProof (FINISHED)
1106.81/291.58	Propagated lower bound.
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(11)
1106.81/291.58	BOUNDS(n^1, INF)
1106.81/291.58	
1106.81/291.58	----------------------------------------
1106.81/291.58	
1106.81/291.58	(12)
1106.81/291.58	Obligation:
1106.81/291.58	TRS:
1106.81/291.58	Rules:
1106.81/291.58	a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
1106.81/291.58	a____(X, nil) -> mark(X)
1106.81/291.58	a____(nil, X) -> mark(X)
1106.81/291.58	a__and(tt, X) -> mark(X)
1106.81/291.58	a__isNePal(__(I, __(P, I))) -> tt
1106.81/291.58	mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
1106.81/291.58	mark(and(X1, X2)) -> a__and(mark(X1), X2)
1106.81/291.58	mark(isNePal(X)) -> a__isNePal(mark(X))
1106.81/291.58	mark(nil) -> nil
1106.81/291.58	mark(tt) -> tt
1106.81/291.58	a____(X1, X2) -> __(X1, X2)
1106.81/291.58	a__and(X1, X2) -> and(X1, X2)
1106.81/291.58	a__isNePal(X) -> isNePal(X)
1106.81/291.58	
1106.81/291.58	Types:
1106.81/291.58	a____ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	__ :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	mark :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	nil :: __:nil:tt:and:isNePal
1106.81/291.58	a__and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	tt :: __:nil:tt:and:isNePal
1106.81/291.58	a__isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	and :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	isNePal :: __:nil:tt:and:isNePal -> __:nil:tt:and:isNePal
1106.81/291.58	hole___:nil:tt:and:isNePal1_0 :: __:nil:tt:and:isNePal
1106.81/291.58	gen___:nil:tt:and:isNePal2_0 :: Nat -> __:nil:tt:and:isNePal
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	Lemmas:
1106.81/291.58	mark(gen___:nil:tt:and:isNePal2_0(n4_0)) -> gen___:nil:tt:and:isNePal2_0(0), rt in Omega(1 + n4_0)
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	Generator Equations:
1106.81/291.58	gen___:nil:tt:and:isNePal2_0(0) <=> nil
1106.81/291.58	gen___:nil:tt:and:isNePal2_0(+(x, 1)) <=> __(nil, gen___:nil:tt:and:isNePal2_0(x))
1106.81/291.58	
1106.81/291.58	
1106.81/291.58	The following defined symbols remain to be analysed:
1106.81/291.58	a____
1106.81/291.58	
1106.81/291.58	They will be analysed ascendingly in the following order:
1106.81/291.58	a____ = mark
1107.07/291.68	EOF