3.37/1.66	YES
3.37/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.37/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.37/1.67	
3.37/1.67	
3.37/1.67	Termination w.r.t. Q of the given QTRS could be proven:
3.37/1.67	
3.37/1.67	(0) QTRS
3.37/1.67	(1) QTRSRRRProof [EQUIVALENT, 25 ms]
3.37/1.67	(2) QTRS
3.37/1.67	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.37/1.67	(4) YES
3.37/1.67	
3.37/1.67	
3.37/1.67	----------------------------------------
3.37/1.67	
3.37/1.67	(0)
3.37/1.67	Obligation:
3.37/1.67	Q restricted rewrite system:
3.37/1.67	The TRS R consists of the following rules:
3.37/1.67	
3.37/1.67	   a__U11(tt, M, N) -> a__U12(tt, M, N)
3.37/1.67	   a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
3.37/1.67	   a__plus(N, 0) -> mark(N)
3.37/1.67	   a__plus(N, s(M)) -> a__U11(tt, M, N)
3.37/1.67	   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
3.37/1.67	   mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
3.37/1.67	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.37/1.67	   mark(tt) -> tt
3.37/1.67	   mark(s(X)) -> s(mark(X))
3.37/1.67	   mark(0) -> 0
3.37/1.67	   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
3.37/1.67	   a__U12(X1, X2, X3) -> U12(X1, X2, X3)
3.37/1.67	   a__plus(X1, X2) -> plus(X1, X2)
3.37/1.67	
3.37/1.67	The set Q consists of the following terms:
3.37/1.67	
3.37/1.67	   mark(U11(x0, x1, x2))
3.37/1.67	   mark(U12(x0, x1, x2))
3.37/1.67	   mark(plus(x0, x1))
3.37/1.67	   mark(tt)
3.37/1.67	   mark(s(x0))
3.37/1.67	   mark(0)
3.37/1.67	   a__U11(x0, x1, x2)
3.37/1.67	   a__U12(x0, x1, x2)
3.37/1.67	   a__plus(x0, x1)
3.37/1.67	
3.37/1.67	
3.37/1.67	----------------------------------------
3.37/1.67	
3.37/1.67	(1) QTRSRRRProof (EQUIVALENT)
3.37/1.67	Used ordering:
3.37/1.67	Knuth-Bendix order [KBO] with precedence:mark_1 > a__plus_2 > plus_2 > tt > a__U11_3 > a__U12_3 > U12_3 > U11_3 > s_1 > 0
3.37/1.67	
3.37/1.67	and weight map:
3.37/1.67	
3.37/1.67	   tt=2
3.37/1.67	   0=1
3.37/1.67	   s_1=2
3.37/1.67	   mark_1=0
3.37/1.67	   a__U11_3=0
3.37/1.67	   a__U12_3=0
3.37/1.67	   a__plus_2=0
3.37/1.67	   U11_3=0
3.37/1.67	   U12_3=0
3.37/1.67	   plus_2=0
3.37/1.67	
3.37/1.67	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.37/1.67	
3.37/1.67	   a__U11(tt, M, N) -> a__U12(tt, M, N)
3.37/1.67	   a__U12(tt, M, N) -> s(a__plus(mark(N), mark(M)))
3.37/1.67	   a__plus(N, 0) -> mark(N)
3.37/1.67	   a__plus(N, s(M)) -> a__U11(tt, M, N)
3.37/1.67	   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
3.37/1.67	   mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3)
3.37/1.67	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
3.37/1.67	   mark(tt) -> tt
3.37/1.67	   mark(s(X)) -> s(mark(X))
3.37/1.67	   mark(0) -> 0
3.37/1.67	   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
3.37/1.67	   a__U12(X1, X2, X3) -> U12(X1, X2, X3)
3.37/1.67	   a__plus(X1, X2) -> plus(X1, X2)
3.37/1.67	
3.37/1.67	
3.37/1.67	
3.37/1.67	
3.37/1.67	----------------------------------------
3.37/1.67	
3.37/1.67	(2)
3.37/1.67	Obligation:
3.37/1.67	Q restricted rewrite system:
3.37/1.67	R is empty.
3.37/1.67	The set Q consists of the following terms:
3.37/1.67	
3.37/1.67	   mark(U11(x0, x1, x2))
3.37/1.67	   mark(U12(x0, x1, x2))
3.37/1.67	   mark(plus(x0, x1))
3.37/1.67	   mark(tt)
3.37/1.67	   mark(s(x0))
3.37/1.67	   mark(0)
3.37/1.67	   a__U11(x0, x1, x2)
3.37/1.67	   a__U12(x0, x1, x2)
3.37/1.67	   a__plus(x0, x1)
3.37/1.67	
3.37/1.67	
3.37/1.67	----------------------------------------
3.37/1.67	
3.37/1.67	(3) RisEmptyProof (EQUIVALENT)
3.37/1.67	The TRS R is empty. Hence, termination is trivially proven.
3.37/1.67	----------------------------------------
3.37/1.67	
3.37/1.67	(4)
3.37/1.67	YES
3.56/1.71	EOF