3.67/1.63	YES
3.67/1.64	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.67/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.67/1.64	
3.67/1.64	
3.67/1.64	Termination of the given CSR could be proven:
3.67/1.64	
3.67/1.64	(0) CSR
3.67/1.64	(1) CSRRRRProof [EQUIVALENT, 55 ms]
3.67/1.64	(2) CSR
3.67/1.64	(3) CSRRRRProof [EQUIVALENT, 10 ms]
3.67/1.64	(4) CSR
3.67/1.64	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.67/1.64	(6) CSR
3.67/1.64	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.67/1.64	(8) CSR
3.67/1.64	(9) CSRRRRProof [EQUIVALENT, 4 ms]
3.67/1.64	(10) CSR
3.67/1.64	(11) CSRRRRProof [EQUIVALENT, 0 ms]
3.67/1.64	(12) CSR
3.67/1.64	(13) CSRRRRProof [EQUIVALENT, 5 ms]
3.67/1.64	(14) CSR
3.67/1.64	(15) RisEmptyProof [EQUIVALENT, 0 ms]
3.67/1.64	(16) YES
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(0)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U31(tt, N) -> N
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	   plus(N, 0) -> U31(isNat(N), N)
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U31: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(1) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U31(tt, N) -> N
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	   plus(N, 0) -> U31(isNat(N), N)
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U31: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(0) = 1
3.67/1.64	   POL(U11(x_1, x_2)) = 2*x_1
3.67/1.64	   POL(U12(x_1)) = 2*x_1
3.67/1.64	   POL(U21(x_1)) = 2*x_1
3.67/1.64	   POL(U31(x_1, x_2)) = 2*x_1 + 2*x_2
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3
3.67/1.64	   POL(isNat(x_1)) = 0
3.67/1.64	   POL(plus(x_1, x_2)) = 2*x_1 + x_2
3.67/1.64	   POL(s(x_1)) = x_1
3.67/1.64	   POL(tt) = 0
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   plus(N, 0) -> U31(isNat(N), N)
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(2)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U31(tt, N) -> N
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U31: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(3) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U31(tt, N) -> N
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U31: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(0) = 0
3.67/1.64	   POL(U11(x_1, x_2)) = x_1
3.67/1.64	   POL(U12(x_1)) = x_1
3.67/1.64	   POL(U21(x_1)) = x_1
3.67/1.64	   POL(U31(x_1, x_2)) = x_1 + x_2
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.67/1.64	   POL(isNat(x_1)) = 1
3.67/1.64	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
3.67/1.64	   POL(s(x_1)) = x_1
3.67/1.64	   POL(tt) = 1
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   U31(tt, N) -> N
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(4)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(5) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(0) = 2
3.67/1.64	   POL(U11(x_1, x_2)) = x_1
3.67/1.64	   POL(U12(x_1)) = 2*x_1
3.67/1.64	   POL(U21(x_1)) = x_1
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
3.67/1.64	   POL(isNat(x_1)) = 0
3.67/1.64	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
3.67/1.64	   POL(s(x_1)) = 2 + x_1
3.67/1.64	   POL(tt) = 0
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   plus(N, s(M)) -> U41(isNat(M), M, N)
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(6)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(7) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U21: {1}
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	0: empty set
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(0) = 1
3.67/1.64	   POL(U11(x_1, x_2)) = x_1 + 2*x_2
3.67/1.64	   POL(U12(x_1)) = 2*x_1
3.67/1.64	   POL(U21(x_1)) = 1 + x_1
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
3.67/1.64	   POL(isNat(x_1)) = x_1
3.67/1.64	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
3.67/1.64	   POL(s(x_1)) = 2 + x_1
3.67/1.64	   POL(tt) = 0
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   U21(tt) -> tt
3.67/1.64	   isNat(0) -> tt
3.67/1.64	   isNat(s(V1)) -> U21(isNat(V1))
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(8)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(9) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U12(tt) -> tt
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(U11(x_1, x_2)) = x_1
3.67/1.64	   POL(U12(x_1)) = 1 + x_1
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
3.67/1.64	   POL(isNat(x_1)) = 0
3.67/1.64	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
3.67/1.64	   POL(s(x_1)) = 1 + x_1
3.67/1.64	   POL(tt) = 1
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   U12(tt) -> tt
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(10)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(11) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	s: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(U11(x_1, x_2)) = x_1
3.67/1.64	   POL(U12(x_1)) = x_1
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.67/1.64	   POL(isNat(x_1)) = 1
3.67/1.64	   POL(plus(x_1, x_2)) = x_1 + x_2
3.67/1.64	   POL(s(x_1)) = x_1
3.67/1.64	   POL(tt) = 1
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   U42(tt, M, N) -> s(plus(N, M))
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(12)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(13) CSRRRRProof (EQUIVALENT)
3.67/1.64	The following CSR is given: Context-sensitive rewrite system:
3.67/1.64	The TRS R consists of the following rules:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	The replacement map contains the following entries:
3.67/1.64	
3.67/1.64	U11: {1}
3.67/1.64	tt: empty set
3.67/1.64	U12: {1}
3.67/1.64	isNat: empty set
3.67/1.64	U41: {1}
3.67/1.64	U42: {1}
3.67/1.64	plus: {1, 2}
3.67/1.64	Used ordering:
3.67/1.64	Polynomial interpretation [POLO]:
3.67/1.64	
3.67/1.64	   POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2
3.67/1.64	   POL(U12(x_1)) = 2*x_1
3.67/1.64	   POL(U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
3.67/1.64	   POL(U42(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
3.67/1.64	   POL(isNat(x_1)) = x_1
3.67/1.64	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.67/1.64	   POL(tt) = 0
3.67/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.67/1.64	
3.67/1.64	   U11(tt, V2) -> U12(isNat(V2))
3.67/1.64	   U41(tt, M, N) -> U42(isNat(N), M, N)
3.67/1.64	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(14)
3.67/1.64	Obligation:
3.67/1.64	Context-sensitive rewrite system:
3.67/1.64	R is empty.
3.67/1.64	
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(15) RisEmptyProof (EQUIVALENT)
3.67/1.64	The CSR R is empty. Hence, termination is trivially proven.
3.67/1.64	----------------------------------------
3.67/1.64	
3.67/1.64	(16)
3.67/1.64	YES
3.67/1.66	EOF