4.05/1.80	YES
4.05/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.05/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.05/1.81	
4.05/1.81	
4.05/1.81	Termination w.r.t. Q of the given QTRS could be proven:
4.05/1.81	
4.05/1.81	(0) QTRS
4.05/1.81	(1) QTRSRRRProof [EQUIVALENT, 132 ms]
4.05/1.81	(2) QTRS
4.05/1.81	(3) QTRSRRRProof [EQUIVALENT, 2 ms]
4.05/1.81	(4) QTRS
4.05/1.81	(5) RisEmptyProof [EQUIVALENT, 0 ms]
4.05/1.81	(6) YES
4.05/1.81	
4.05/1.81	
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(0)
4.05/1.81	Obligation:
4.05/1.81	Q restricted rewrite system:
4.05/1.81	The TRS R consists of the following rules:
4.05/1.81	
4.05/1.81	   a__U11(tt, N) -> mark(N)
4.05/1.81	   a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M)))
4.05/1.81	   a__and(tt, X) -> mark(X)
4.05/1.81	   a__isNat(0) -> tt
4.05/1.81	   a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
4.05/1.81	   a__isNat(s(V1)) -> a__isNat(V1)
4.05/1.81	   a__plus(N, 0) -> a__U11(a__isNat(N), N)
4.05/1.81	   a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N)
4.05/1.81	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
4.05/1.81	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
4.05/1.81	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
4.05/1.81	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
4.05/1.81	   mark(isNat(X)) -> a__isNat(X)
4.05/1.81	   mark(tt) -> tt
4.05/1.81	   mark(s(X)) -> s(mark(X))
4.05/1.81	   mark(0) -> 0
4.05/1.81	   a__U11(X1, X2) -> U11(X1, X2)
4.05/1.81	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
4.05/1.81	   a__plus(X1, X2) -> plus(X1, X2)
4.05/1.81	   a__and(X1, X2) -> and(X1, X2)
4.05/1.81	   a__isNat(X) -> isNat(X)
4.05/1.81	
4.05/1.81	The set Q consists of the following terms:
4.05/1.81	
4.05/1.81	   mark(U11(x0, x1))
4.05/1.81	   mark(U21(x0, x1, x2))
4.05/1.81	   mark(plus(x0, x1))
4.05/1.81	   mark(and(x0, x1))
4.05/1.81	   mark(isNat(x0))
4.05/1.81	   mark(tt)
4.05/1.81	   mark(s(x0))
4.05/1.81	   mark(0)
4.05/1.81	   a__U11(x0, x1)
4.05/1.81	   a__U21(x0, x1, x2)
4.05/1.81	   a__plus(x0, x1)
4.05/1.81	   a__and(x0, x1)
4.05/1.81	   a__isNat(x0)
4.05/1.81	
4.05/1.81	
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(1) QTRSRRRProof (EQUIVALENT)
4.05/1.81	Used ordering:
4.05/1.81	a__U11/2(YES,YES)
4.05/1.81	tt/0)
4.05/1.81	mark/1)YES(
4.05/1.81	a__U21/3(YES,YES,YES)
4.05/1.81	s/1(YES)
4.05/1.81	a__plus/2(YES,YES)
4.05/1.81	a__and/2(YES,YES)
4.05/1.81	a__isNat/1(YES)
4.05/1.81	0/0)
4.05/1.81	plus/2(YES,YES)
4.05/1.81	isNat/1(YES)
4.05/1.81	U11/2(YES,YES)
4.05/1.81	U21/3(YES,YES,YES)
4.05/1.81	and/2(YES,YES)
4.05/1.81	
4.05/1.81	Quasi precedence:
4.05/1.81	0 > tt > [a__U21_3, a__plus_2, plus_2, U21_3] > [a__U11_2, U11_2] > s_1
4.05/1.81	0 > tt > [a__U21_3, a__plus_2, plus_2, U21_3] > [a__and_2, a__isNat_1, isNat_1, and_2] > s_1
4.05/1.81	
4.05/1.81	
4.05/1.81	Status:
4.05/1.81	a__U11_2: [2,1]
4.05/1.81	tt: multiset status
4.05/1.81	a__U21_3: [3,2,1]
4.05/1.81	s_1: multiset status
4.05/1.81	a__plus_2: [1,2]
4.05/1.81	a__and_2: multiset status
4.05/1.81	a__isNat_1: multiset status
4.05/1.81	0: multiset status
4.05/1.81	plus_2: [1,2]
4.05/1.81	isNat_1: multiset status
4.05/1.81	U11_2: [2,1]
4.05/1.81	U21_3: [3,2,1]
4.05/1.81	and_2: multiset status
4.05/1.81	
4.05/1.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.05/1.81	
4.05/1.81	   a__U11(tt, N) -> mark(N)
4.05/1.81	   a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M)))
4.05/1.81	   a__and(tt, X) -> mark(X)
4.05/1.81	   a__isNat(0) -> tt
4.05/1.81	   a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2))
4.05/1.81	   a__isNat(s(V1)) -> a__isNat(V1)
4.05/1.81	   a__plus(N, 0) -> a__U11(a__isNat(N), N)
4.05/1.81	   a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N)
4.05/1.81	
4.05/1.81	
4.05/1.81	
4.05/1.81	
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(2)
4.05/1.81	Obligation:
4.05/1.81	Q restricted rewrite system:
4.05/1.81	The TRS R consists of the following rules:
4.05/1.81	
4.05/1.81	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
4.05/1.81	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
4.05/1.81	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
4.05/1.81	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
4.05/1.81	   mark(isNat(X)) -> a__isNat(X)
4.05/1.81	   mark(tt) -> tt
4.05/1.81	   mark(s(X)) -> s(mark(X))
4.05/1.81	   mark(0) -> 0
4.05/1.81	   a__U11(X1, X2) -> U11(X1, X2)
4.05/1.81	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
4.05/1.81	   a__plus(X1, X2) -> plus(X1, X2)
4.05/1.81	   a__and(X1, X2) -> and(X1, X2)
4.05/1.81	   a__isNat(X) -> isNat(X)
4.05/1.81	
4.05/1.81	The set Q consists of the following terms:
4.05/1.81	
4.05/1.81	   mark(U11(x0, x1))
4.05/1.81	   mark(U21(x0, x1, x2))
4.05/1.81	   mark(plus(x0, x1))
4.05/1.81	   mark(and(x0, x1))
4.05/1.81	   mark(isNat(x0))
4.05/1.81	   mark(tt)
4.05/1.81	   mark(s(x0))
4.05/1.81	   mark(0)
4.05/1.81	   a__U11(x0, x1)
4.05/1.81	   a__U21(x0, x1, x2)
4.05/1.81	   a__plus(x0, x1)
4.05/1.81	   a__and(x0, x1)
4.05/1.81	   a__isNat(x0)
4.05/1.81	
4.05/1.81	
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(3) QTRSRRRProof (EQUIVALENT)
4.05/1.81	Used ordering:
4.05/1.81	Knuth-Bendix order [KBO] with precedence:mark_1 > 0 > s_1 > a__and_2 > and_2 > tt > a__U21_3 > U21_3 > a__isNat_1 > a__U11_2 > isNat_1 > a__plus_2 > U11_2 > plus_2
4.05/1.81	
4.05/1.81	and weight map:
4.05/1.81	
4.05/1.81	   tt=1
4.05/1.81	   0=1
4.05/1.81	   mark_1=0
4.05/1.81	   isNat_1=1
4.05/1.81	   a__isNat_1=1
4.05/1.81	   s_1=1
4.05/1.81	   U11_2=0
4.05/1.81	   a__U11_2=0
4.05/1.81	   U21_3=0
4.05/1.81	   a__U21_3=0
4.05/1.81	   plus_2=0
4.05/1.81	   a__plus_2=0
4.05/1.81	   and_2=0
4.05/1.81	   a__and_2=0
4.05/1.81	
4.05/1.81	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.05/1.81	
4.05/1.81	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
4.05/1.81	   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
4.05/1.81	   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
4.05/1.81	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
4.05/1.81	   mark(isNat(X)) -> a__isNat(X)
4.05/1.81	   mark(tt) -> tt
4.05/1.81	   mark(s(X)) -> s(mark(X))
4.05/1.81	   mark(0) -> 0
4.05/1.81	   a__U11(X1, X2) -> U11(X1, X2)
4.05/1.81	   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
4.05/1.81	   a__plus(X1, X2) -> plus(X1, X2)
4.05/1.81	   a__and(X1, X2) -> and(X1, X2)
4.05/1.81	   a__isNat(X) -> isNat(X)
4.05/1.81	
4.05/1.81	
4.05/1.81	
4.05/1.81	
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(4)
4.05/1.81	Obligation:
4.05/1.81	Q restricted rewrite system:
4.05/1.81	R is empty.
4.05/1.81	The set Q consists of the following terms:
4.05/1.81	
4.05/1.81	   mark(U11(x0, x1))
4.05/1.81	   mark(U21(x0, x1, x2))
4.05/1.81	   mark(plus(x0, x1))
4.05/1.81	   mark(and(x0, x1))
4.05/1.81	   mark(isNat(x0))
4.05/1.81	   mark(tt)
4.05/1.81	   mark(s(x0))
4.05/1.81	   mark(0)
4.05/1.81	   a__U11(x0, x1)
4.05/1.81	   a__U21(x0, x1, x2)
4.05/1.81	   a__plus(x0, x1)
4.05/1.81	   a__and(x0, x1)
4.05/1.81	   a__isNat(x0)
4.05/1.81	
4.05/1.81	
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(5) RisEmptyProof (EQUIVALENT)
4.05/1.81	The TRS R is empty. Hence, termination is trivially proven.
4.05/1.81	----------------------------------------
4.05/1.81	
4.05/1.81	(6)
4.05/1.81	YES
4.22/1.83	EOF