5.15/2.03	YES
5.37/2.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.37/2.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.37/2.05	
5.37/2.05	
5.37/2.05	Termination of the given CSR could be proven:
5.37/2.05	
5.37/2.05	(0) CSR
5.37/2.05	(1) CSRRRRProof [EQUIVALENT, 96 ms]
5.37/2.05	(2) CSR
5.37/2.05	(3) CSRRRRProof [EQUIVALENT, 18 ms]
5.37/2.05	(4) CSR
5.37/2.05	(5) CSRRRRProof [EQUIVALENT, 0 ms]
5.37/2.05	(6) CSR
5.37/2.05	(7) CSRRRRProof [EQUIVALENT, 0 ms]
5.37/2.05	(8) CSR
5.37/2.05	(9) CSRRRRProof [EQUIVALENT, 11 ms]
5.37/2.05	(10) CSR
5.37/2.05	(11) CSRRRRProof [EQUIVALENT, 11 ms]
5.37/2.05	(12) CSR
5.37/2.05	(13) CSRRRRProof [EQUIVALENT, 0 ms]
5.37/2.05	(14) CSR
5.37/2.05	(15) CSRRRRProof [EQUIVALENT, 11 ms]
5.37/2.05	(16) CSR
5.37/2.05	(17) CSRRRRProof [EQUIVALENT, 0 ms]
5.37/2.05	(18) CSR
5.37/2.05	(19) CSRRRRProof [EQUIVALENT, 0 ms]
5.37/2.05	(20) CSR
5.37/2.05	(21) RisEmptyProof [EQUIVALENT, 0 ms]
5.37/2.05	(22) YES
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(0)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U51(tt, N) -> U52(isNatKind(N), N)
5.37/2.05	   U52(tt, N) -> N
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	   plus(N, 0) -> U51(isNat(N), N)
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U51: {1}
5.37/2.05	U52: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(1) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U51(tt, N) -> U52(isNatKind(N), N)
5.37/2.05	   U52(tt, N) -> N
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	   plus(N, 0) -> U51(isNat(N), N)
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U51: {1}
5.37/2.05	U52: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 2
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = 2*x_1
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = 2*x_1
5.37/2.05	   POL(U15(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = x_1
5.37/2.05	   POL(U22(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = 2*x_1
5.37/2.05	   POL(U41(x_1)) = 2*x_1
5.37/2.05	   POL(U51(x_1, x_2)) = 1 + 2*x_1 + x_2
5.37/2.05	   POL(U52(x_1, x_2)) = x_1 + x_2
5.37/2.05	   POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
5.37/2.05	   POL(U62(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = 0
5.37/2.05	   POL(isNatKind(x_1)) = 0
5.37/2.05	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
5.37/2.05	   POL(s(x_1)) = x_1
5.37/2.05	   POL(tt) = 0
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U51(tt, N) -> U52(isNatKind(N), N)
5.37/2.05	   plus(N, 0) -> U51(isNat(N), N)
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(2)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U52(tt, N) -> N
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U52: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(3) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U52(tt, N) -> N
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U52: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 1
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U15(x_1, x_2)) = x_1
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = x_1
5.37/2.05	   POL(U22(x_1, x_2)) = x_1
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = x_1
5.37/2.05	   POL(U41(x_1)) = x_1
5.37/2.05	   POL(U52(x_1, x_2)) = x_1 + x_2
5.37/2.05	   POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U62(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = 1
5.37/2.05	   POL(isNatKind(x_1)) = 1
5.37/2.05	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(s(x_1)) = 1 + x_1
5.37/2.05	   POL(tt) = 1
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U52(tt, N) -> N
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(4)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(5) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 0
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = 2*x_1
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = 2*x_1
5.37/2.05	   POL(U15(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U22(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U32(x_1)) = 2*x_1
5.37/2.05	   POL(U41(x_1)) = 2*x_1
5.37/2.05	   POL(U61(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
5.37/2.05	   POL(U62(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = 0
5.37/2.05	   POL(isNatKind(x_1)) = 0
5.37/2.05	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
5.37/2.05	   POL(s(x_1)) = 2 + x_1
5.37/2.05	   POL(tt) = 0
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   plus(N, s(M)) -> U61(isNat(M), M, N)
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(6)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(7) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 1
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U15(x_1, x_2)) = x_1
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = x_1
5.37/2.05	   POL(U22(x_1, x_2)) = x_1
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = x_1
5.37/2.05	   POL(U41(x_1)) = x_1
5.37/2.05	   POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U62(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = 1
5.37/2.05	   POL(isNatKind(x_1)) = 1
5.37/2.05	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(s(x_1)) = x_1
5.37/2.05	   POL(tt) = 1
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U62(tt, M, N) -> U63(isNat(N), M, N)
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(8)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(9) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U61: {1}
5.37/2.05	U62: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 1
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U15(x_1, x_2)) = x_1
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = x_1
5.37/2.05	   POL(U22(x_1, x_2)) = x_1
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = x_1
5.37/2.05	   POL(U41(x_1)) = x_1
5.37/2.05	   POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U62(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = 0
5.37/2.05	   POL(isNatKind(x_1)) = 0
5.37/2.05	   POL(plus(x_1, x_2)) = x_1 + x_2
5.37/2.05	   POL(s(x_1)) = x_1
5.37/2.05	   POL(tt) = 0
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
5.37/2.05	   U64(tt, M, N) -> s(plus(N, M))
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(10)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(11) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   isNat(0) -> tt
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 1
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U15(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(U22(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(U23(x_1)) = 1 + x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = x_1
5.37/2.05	   POL(U41(x_1)) = x_1
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = x_1
5.37/2.05	   POL(isNatKind(x_1)) = 0
5.37/2.05	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(s(x_1)) = 1 + x_1
5.37/2.05	   POL(tt) = 0
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U15(tt, V2) -> U16(isNat(V2))
5.37/2.05	   U23(tt) -> tt
5.37/2.05	   isNat(0) -> tt
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(12)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(13) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	U63: {1}
5.37/2.05	U64: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 1
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(U15(x_1, x_2)) = x_1 + x_2
5.37/2.05	   POL(U16(x_1)) = x_1
5.37/2.05	   POL(U21(x_1, x_2)) = x_1 + x_2
5.37/2.05	   POL(U22(x_1, x_2)) = x_1 + x_2
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = x_1
5.37/2.05	   POL(U41(x_1)) = x_1
5.37/2.05	   POL(U63(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U64(x_1, x_2, x_3)) = x_1 + x_2 + x_3
5.37/2.05	   POL(isNat(x_1)) = 1 + x_1
5.37/2.05	   POL(isNatKind(x_1)) = 1
5.37/2.05	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(s(x_1)) = 1 + x_1
5.37/2.05	   POL(tt) = 1
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
5.37/2.05	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.37/2.05	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(14)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(15) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U14: {1}
5.37/2.05	U15: {1}
5.37/2.05	isNat: empty set
5.37/2.05	U16: {1}
5.37/2.05	U21: {1}
5.37/2.05	U22: {1}
5.37/2.05	U23: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 2
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2
5.37/2.05	   POL(U14(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2
5.37/2.05	   POL(U15(x_1, x_2)) = 2*x_1
5.37/2.05	   POL(U16(x_1)) = 1 + x_1
5.37/2.05	   POL(U21(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
5.37/2.05	   POL(U22(x_1, x_2)) = 1 + x_1 + 2*x_2
5.37/2.05	   POL(U23(x_1)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = x_1
5.37/2.05	   POL(U32(x_1)) = x_1
5.37/2.05	   POL(U41(x_1)) = 2*x_1
5.37/2.05	   POL(isNat(x_1)) = x_1
5.37/2.05	   POL(isNatKind(x_1)) = 0
5.37/2.05	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_2
5.37/2.05	   POL(s(x_1)) = 2 + 2*x_1
5.37/2.05	   POL(tt) = 0
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.37/2.05	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.37/2.05	   U16(tt) -> tt
5.37/2.05	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.37/2.05	   U22(tt, V1) -> U23(isNat(V1))
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(16)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(17) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 0
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_3
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = 1 + x_1
5.37/2.05	   POL(U31(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(U32(x_1)) = 1 + x_1
5.37/2.05	   POL(U41(x_1)) = 1 + x_1
5.37/2.05	   POL(isNatKind(x_1)) = 1 + x_1
5.37/2.05	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
5.37/2.05	   POL(s(x_1)) = 1 + x_1
5.37/2.05	   POL(tt) = 1
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U32(tt) -> tt
5.37/2.05	   U41(tt) -> tt
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(18)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(19) CSRRRRProof (EQUIVALENT)
5.37/2.05	The following CSR is given: Context-sensitive rewrite system:
5.37/2.05	The TRS R consists of the following rules:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	The replacement map contains the following entries:
5.37/2.05	
5.37/2.05	U11: {1}
5.37/2.05	tt: empty set
5.37/2.05	U12: {1}
5.37/2.05	isNatKind: empty set
5.37/2.05	U13: {1}
5.37/2.05	U31: {1}
5.37/2.05	U32: {1}
5.37/2.05	U41: {1}
5.37/2.05	s: {1}
5.37/2.05	plus: {1, 2}
5.37/2.05	0: empty set
5.37/2.05	Used ordering:
5.37/2.05	Polynomial interpretation [POLO]:
5.37/2.05	
5.37/2.05	   POL(0) = 2
5.37/2.05	   POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
5.37/2.05	   POL(U12(x_1, x_2, x_3)) = x_1 + 2*x_3
5.37/2.05	   POL(U13(x_1, x_2, x_3)) = x_1
5.37/2.05	   POL(U31(x_1, x_2)) = 2*x_1 + 2*x_2
5.37/2.05	   POL(U32(x_1)) = 2 + x_1
5.37/2.05	   POL(U41(x_1)) = 1 + x_1
5.37/2.05	   POL(isNatKind(x_1)) = 1 + 2*x_1
5.37/2.05	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_2
5.37/2.05	   POL(s(x_1)) = 2 + 2*x_1
5.37/2.05	   POL(tt) = 2
5.37/2.05	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.37/2.05	
5.37/2.05	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.37/2.05	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.37/2.05	   U31(tt, V2) -> U32(isNatKind(V2))
5.37/2.05	   isNatKind(0) -> tt
5.37/2.05	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
5.37/2.05	   isNatKind(s(V1)) -> U41(isNatKind(V1))
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(20)
5.37/2.05	Obligation:
5.37/2.05	Context-sensitive rewrite system:
5.37/2.05	R is empty.
5.37/2.05	
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(21) RisEmptyProof (EQUIVALENT)
5.37/2.05	The CSR R is empty. Hence, termination is trivially proven.
5.37/2.05	----------------------------------------
5.37/2.05	
5.37/2.05	(22)
5.37/2.05	YES
5.41/2.09	EOF