3.29/1.67	YES
3.29/1.68	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.29/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.29/1.68	
3.29/1.68	
3.29/1.68	Termination w.r.t. Q of the given QTRS could be proven:
3.29/1.68	
3.29/1.68	(0) QTRS
3.29/1.68	(1) QTRSRRRProof [EQUIVALENT, 32 ms]
3.29/1.68	(2) QTRS
3.29/1.68	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.29/1.68	(4) YES
3.29/1.68	
3.29/1.68	
3.29/1.68	----------------------------------------
3.29/1.68	
3.29/1.68	(0)
3.29/1.68	Obligation:
3.29/1.68	Q restricted rewrite system:
3.29/1.68	The TRS R consists of the following rules:
3.29/1.68	
3.29/1.68	   a__and(tt, X) -> mark(X)
3.29/1.68	   a__plus(N, 0) -> mark(N)
3.29/1.68	   a__plus(N, s(M)) -> s(a__plus(mark(N), mark(M)))
3.29/1.68	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
3.29/1.68	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.29/1.68	   mark(tt) -> tt
3.29/1.68	   mark(0) -> 0
3.29/1.68	   mark(s(X)) -> s(mark(X))
3.29/1.68	   a__and(X1, X2) -> and(X1, X2)
3.29/1.68	   a__plus(X1, X2) -> plus(X1, X2)
3.29/1.68	
3.29/1.68	The set Q consists of the following terms:
3.29/1.68	
3.29/1.68	   mark(and(x0, x1))
3.29/1.68	   mark(plus(x0, x1))
3.29/1.68	   mark(tt)
3.29/1.68	   mark(0)
3.29/1.68	   mark(s(x0))
3.29/1.68	   a__and(x0, x1)
3.29/1.68	   a__plus(x0, x1)
3.29/1.68	
3.29/1.68	
3.29/1.68	----------------------------------------
3.29/1.68	
3.29/1.68	(1) QTRSRRRProof (EQUIVALENT)
3.29/1.68	Used ordering:
3.29/1.68	Knuth-Bendix order [KBO] with precedence:mark_1 > a__and_2 > and_2 > a__plus_2 > plus_2 > s_1 > 0 > tt
3.29/1.68	
3.29/1.68	and weight map:
3.29/1.68	
3.29/1.68	   tt=1
3.29/1.68	   0=1
3.29/1.68	   mark_1=0
3.29/1.68	   s_1=1
3.29/1.68	   a__and_2=0
3.29/1.68	   a__plus_2=0
3.29/1.68	   and_2=0
3.29/1.68	   plus_2=0
3.29/1.68	
3.29/1.68	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.29/1.68	
3.29/1.68	   a__and(tt, X) -> mark(X)
3.29/1.68	   a__plus(N, 0) -> mark(N)
3.29/1.68	   a__plus(N, s(M)) -> s(a__plus(mark(N), mark(M)))
3.29/1.68	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
3.29/1.68	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.29/1.68	   mark(tt) -> tt
3.29/1.68	   mark(0) -> 0
3.29/1.68	   mark(s(X)) -> s(mark(X))
3.29/1.68	   a__and(X1, X2) -> and(X1, X2)
3.29/1.68	   a__plus(X1, X2) -> plus(X1, X2)
3.29/1.68	
3.29/1.68	
3.29/1.68	
3.29/1.68	
3.29/1.68	----------------------------------------
3.29/1.68	
3.29/1.68	(2)
3.29/1.68	Obligation:
3.29/1.68	Q restricted rewrite system:
3.29/1.68	R is empty.
3.29/1.68	The set Q consists of the following terms:
3.29/1.68	
3.29/1.68	   mark(and(x0, x1))
3.29/1.68	   mark(plus(x0, x1))
3.29/1.68	   mark(tt)
3.29/1.68	   mark(0)
3.29/1.68	   mark(s(x0))
3.29/1.68	   a__and(x0, x1)
3.29/1.68	   a__plus(x0, x1)
3.29/1.68	
3.29/1.68	
3.29/1.68	----------------------------------------
3.29/1.68	
3.29/1.68	(3) RisEmptyProof (EQUIVALENT)
3.29/1.68	The TRS R is empty. Hence, termination is trivially proven.
3.29/1.68	----------------------------------------
3.29/1.68	
3.29/1.68	(4)
3.29/1.68	YES
3.49/1.71	EOF