3.60/1.56	YES
3.60/1.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.60/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/1.57	
3.60/1.57	
3.60/1.57	Termination of the given CSR could be proven:
3.60/1.57	
3.60/1.57	(0) CSR
3.60/1.57	(1) CSRRRRProof [EQUIVALENT, 31 ms]
3.60/1.57	(2) CSR
3.60/1.57	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.60/1.57	(4) CSR
3.60/1.57	(5) CSRRRRProof [EQUIVALENT, 5 ms]
3.60/1.57	(6) CSR
3.60/1.57	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.60/1.57	(8) CSR
3.60/1.57	(9) CSRRRRProof [EQUIVALENT, 3 ms]
3.60/1.57	(10) CSR
3.60/1.57	(11) CSRRRRProof [EQUIVALENT, 0 ms]
3.60/1.57	(12) CSR
3.60/1.57	(13) RisEmptyProof [EQUIVALENT, 0 ms]
3.60/1.57	(14) YES
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(0)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   U11(tt, N) -> N
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, 0) -> U11(isNat(N), N)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	U11: {1}
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(1) CSRRRRProof (EQUIVALENT)
3.60/1.57	The following CSR is given: Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   U11(tt, N) -> N
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, 0) -> U11(isNat(N), N)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	U11: {1}
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	Used ordering:
3.60/1.57	Polynomial interpretation [POLO]:
3.60/1.57	
3.60/1.57	   POL(0) = 0
3.60/1.57	   POL(U11(x_1, x_2)) = 1 + x_1 + x_2
3.60/1.57	   POL(U21(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
3.60/1.57	   POL(and(x_1, x_2)) = x_1 + x_2
3.60/1.57	   POL(isNat(x_1)) = 0
3.60/1.57	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
3.60/1.57	   POL(s(x_1)) = x_1
3.60/1.57	   POL(tt) = 0
3.60/1.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.60/1.57	
3.60/1.57	   U11(tt, N) -> N
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(2)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, 0) -> U11(isNat(N), N)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	U11: {1}
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(3) CSRRRRProof (EQUIVALENT)
3.60/1.57	The following CSR is given: Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, 0) -> U11(isNat(N), N)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	U11: {1}
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	Used ordering:
3.60/1.57	Polynomial interpretation [POLO]:
3.60/1.57	
3.60/1.57	   POL(0) = 0
3.60/1.57	   POL(U11(x_1, x_2)) = x_1
3.60/1.57	   POL(U21(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
3.60/1.57	   POL(and(x_1, x_2)) = x_1 + x_2
3.60/1.57	   POL(isNat(x_1)) = 0
3.60/1.57	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
3.60/1.57	   POL(s(x_1)) = x_1
3.60/1.57	   POL(tt) = 0
3.60/1.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.60/1.57	
3.60/1.57	   plus(N, 0) -> U11(isNat(N), N)
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(4)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(5) CSRRRRProof (EQUIVALENT)
3.60/1.57	The following CSR is given: Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	Used ordering:
3.60/1.57	Polynomial interpretation [POLO]:
3.60/1.57	
3.60/1.57	   POL(0) = 2
3.60/1.57	   POL(U21(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
3.60/1.57	   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
3.60/1.57	   POL(isNat(x_1)) = 0
3.60/1.57	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
3.60/1.57	   POL(s(x_1)) = 1 + x_1
3.60/1.57	   POL(tt) = 0
3.60/1.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.60/1.57	
3.60/1.57	   U21(tt, M, N) -> s(plus(N, M))
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(6)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(7) CSRRRRProof (EQUIVALENT)
3.60/1.57	The following CSR is given: Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	U21: {1}
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	Used ordering:
3.60/1.57	Polynomial interpretation [POLO]:
3.60/1.57	
3.60/1.57	   POL(0) = 2
3.60/1.57	   POL(U21(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
3.60/1.57	   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
3.60/1.57	   POL(isNat(x_1)) = 0
3.60/1.57	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_2
3.60/1.57	   POL(s(x_1)) = 1 + 2*x_1
3.60/1.57	   POL(tt) = 0
3.60/1.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.60/1.57	
3.60/1.57	   plus(N, s(M)) -> U21(and(isNat(M), isNat(N)), M, N)
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(8)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(9) CSRRRRProof (EQUIVALENT)
3.60/1.57	The following CSR is given: Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	s: {1}
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	Used ordering:
3.60/1.57	Polynomial interpretation [POLO]:
3.60/1.57	
3.60/1.57	   POL(0) = 1
3.60/1.57	   POL(and(x_1, x_2)) = 1 + x_1 + x_2
3.60/1.57	   POL(isNat(x_1)) = x_1
3.60/1.57	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
3.60/1.57	   POL(s(x_1)) = 1 + x_1
3.60/1.57	   POL(tt) = 1
3.60/1.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.60/1.57	
3.60/1.57	   and(tt, X) -> X
3.60/1.57	   isNat(s(V1)) -> isNat(V1)
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(10)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(11) CSRRRRProof (EQUIVALENT)
3.60/1.57	The following CSR is given: Context-sensitive rewrite system:
3.60/1.57	The TRS R consists of the following rules:
3.60/1.57	
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	
3.60/1.57	The replacement map contains the following entries:
3.60/1.57	
3.60/1.57	tt: empty set
3.60/1.57	plus: {1, 2}
3.60/1.57	and: {1}
3.60/1.57	isNat: empty set
3.60/1.57	0: empty set
3.60/1.57	Used ordering:
3.60/1.57	Polynomial interpretation [POLO]:
3.60/1.57	
3.60/1.57	   POL(0) = 1
3.60/1.57	   POL(and(x_1, x_2)) = 2*x_1
3.60/1.57	   POL(isNat(x_1)) = 1 + 2*x_1
3.60/1.57	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_2
3.60/1.57	   POL(tt) = 2
3.60/1.57	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.60/1.57	
3.60/1.57	   isNat(0) -> tt
3.60/1.57	   isNat(plus(V1, V2)) -> and(isNat(V1), isNat(V2))
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(12)
3.60/1.57	Obligation:
3.60/1.57	Context-sensitive rewrite system:
3.60/1.57	R is empty.
3.60/1.57	
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(13) RisEmptyProof (EQUIVALENT)
3.60/1.57	The CSR R is empty. Hence, termination is trivially proven.
3.60/1.57	----------------------------------------
3.60/1.57	
3.60/1.57	(14)
3.60/1.57	YES
3.60/1.59	EOF