4.50/1.92	YES
4.50/1.94	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.50/1.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.50/1.94	
4.50/1.94	
4.50/1.94	Termination of the given CSR could be proven:
4.50/1.94	
4.50/1.94	(0) CSR
4.50/1.94	(1) CSRRRRProof [EQUIVALENT, 55 ms]
4.50/1.94	(2) CSR
4.50/1.94	(3) CSRRRRProof [EQUIVALENT, 23 ms]
4.50/1.94	(4) CSR
4.50/1.94	(5) CSRRRRProof [EQUIVALENT, 0 ms]
4.50/1.94	(6) CSR
4.50/1.94	(7) CSRRRRProof [EQUIVALENT, 11 ms]
4.50/1.94	(8) CSR
4.50/1.94	(9) CSRRRRProof [EQUIVALENT, 0 ms]
4.50/1.94	(10) CSR
4.50/1.94	(11) CSRRRRProof [EQUIVALENT, 20 ms]
4.50/1.94	(12) CSR
4.50/1.94	(13) CSRRRRProof [EQUIVALENT, 0 ms]
4.50/1.94	(14) CSR
4.50/1.94	(15) CSRRRRProof [EQUIVALENT, 7 ms]
4.50/1.94	(16) CSR
4.50/1.94	(17) CSRRRRProof [EQUIVALENT, 0 ms]
4.50/1.94	(18) CSR
4.50/1.94	(19) CSRRRRProof [EQUIVALENT, 0 ms]
4.50/1.94	(20) CSR
4.50/1.94	(21) CSRRRRProof [EQUIVALENT, 5 ms]
4.50/1.94	(22) CSR
4.50/1.94	(23) RisEmptyProof [EQUIVALENT, 0 ms]
4.50/1.94	(24) YES
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(0)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(nil) -> tt
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(nil) -> 0
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	nil: empty set
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(1) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(nil) -> tt
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(nil) -> 0
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	nil: empty set
4.50/1.94	Used ordering:
4.50/1.94	Polynomial interpretation [POLO]:
4.50/1.94	
4.50/1.94	   POL(0) = 0
4.50/1.94	   POL(U11(x_1)) = x_1
4.50/1.94	   POL(U21(x_1)) = x_1
4.50/1.94	   POL(U31(x_1)) = x_1
4.50/1.94	   POL(U41(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(U42(x_1)) = x_1
4.50/1.94	   POL(U51(x_1, x_2)) = x_1
4.50/1.94	   POL(U52(x_1)) = x_1
4.50/1.94	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + x_3
4.50/1.94	   POL(U62(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(isNat(x_1)) = 1
4.50/1.94	   POL(isNatIList(x_1)) = 1 + x_1
4.50/1.94	   POL(isNatList(x_1)) = 1
4.50/1.94	   POL(length(x_1)) = 1 + x_1
4.50/1.94	   POL(nil) = 1
4.50/1.94	   POL(s(x_1)) = x_1
4.50/1.94	   POL(tt) = 1
4.50/1.94	   POL(zeros) = 0
4.50/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.94	
4.50/1.94	   length(nil) -> 0
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(2)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(nil) -> tt
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	nil: empty set
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(3) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(nil) -> tt
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	nil: empty set
4.50/1.94	Used ordering:
4.50/1.94	Polynomial interpretation [POLO]:
4.50/1.94	
4.50/1.94	   POL(0) = 0
4.50/1.94	   POL(U11(x_1)) = x_1
4.50/1.94	   POL(U21(x_1)) = x_1
4.50/1.94	   POL(U31(x_1)) = x_1
4.50/1.94	   POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2
4.50/1.94	   POL(U42(x_1)) = 2*x_1
4.50/1.94	   POL(U51(x_1, x_2)) = x_1 + 2*x_2
4.50/1.94	   POL(U52(x_1)) = x_1
4.50/1.94	   POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
4.50/1.94	   POL(U62(x_1, x_2)) = x_1 + 2*x_2
4.50/1.94	   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
4.50/1.94	   POL(isNat(x_1)) = x_1
4.50/1.94	   POL(isNatIList(x_1)) = x_1
4.50/1.94	   POL(isNatList(x_1)) = x_1
4.50/1.94	   POL(length(x_1)) = 2*x_1
4.50/1.94	   POL(nil) = 2
4.50/1.94	   POL(s(x_1)) = x_1
4.50/1.94	   POL(tt) = 0
4.50/1.94	   POL(zeros) = 0
4.50/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.94	
4.50/1.94	   isNatList(nil) -> tt
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(4)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(5) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	Used ordering:
4.50/1.94	Polynomial interpretation [POLO]:
4.50/1.94	
4.50/1.94	   POL(0) = 0
4.50/1.94	   POL(U11(x_1)) = x_1
4.50/1.94	   POL(U21(x_1)) = x_1
4.50/1.94	   POL(U31(x_1)) = x_1
4.50/1.94	   POL(U41(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(U42(x_1)) = x_1
4.50/1.94	   POL(U51(x_1, x_2)) = x_1
4.50/1.94	   POL(U52(x_1)) = x_1
4.50/1.94	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
4.50/1.94	   POL(U62(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(isNat(x_1)) = 1
4.50/1.94	   POL(isNatIList(x_1)) = 1 + x_1
4.50/1.94	   POL(isNatList(x_1)) = 1
4.50/1.94	   POL(length(x_1)) = 1 + x_1
4.50/1.94	   POL(s(x_1)) = x_1
4.50/1.94	   POL(tt) = 1
4.50/1.94	   POL(zeros) = 1
4.50/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.94	
4.50/1.94	   isNatIList(zeros) -> tt
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(6)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(7) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U31(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	Used ordering:
4.50/1.94	Polynomial interpretation [POLO]:
4.50/1.94	
4.50/1.94	   POL(0) = 0
4.50/1.94	   POL(U11(x_1)) = x_1
4.50/1.94	   POL(U21(x_1)) = x_1
4.50/1.94	   POL(U31(x_1)) = 1 + x_1
4.50/1.94	   POL(U41(x_1, x_2)) = 1 + x_1 + x_2
4.50/1.94	   POL(U42(x_1)) = x_1
4.50/1.94	   POL(U51(x_1, x_2)) = x_1
4.50/1.94	   POL(U52(x_1)) = x_1
4.50/1.94	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
4.50/1.94	   POL(U62(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(isNat(x_1)) = 0
4.50/1.94	   POL(isNatIList(x_1)) = 1 + x_1
4.50/1.94	   POL(isNatList(x_1)) = 0
4.50/1.94	   POL(length(x_1)) = x_1
4.50/1.94	   POL(s(x_1)) = x_1
4.50/1.94	   POL(tt) = 0
4.50/1.94	   POL(zeros) = 1
4.50/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.94	
4.50/1.94	   U31(tt) -> tt
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(8)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(9) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U31: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	Used ordering:
4.50/1.94	Polynomial interpretation [POLO]:
4.50/1.94	
4.50/1.94	   POL(0) = 0
4.50/1.94	   POL(U11(x_1)) = x_1
4.50/1.94	   POL(U21(x_1)) = x_1
4.50/1.94	   POL(U31(x_1)) = x_1
4.50/1.94	   POL(U41(x_1, x_2)) = 1 + x_1 + x_2
4.50/1.94	   POL(U42(x_1)) = x_1
4.50/1.94	   POL(U51(x_1, x_2)) = x_1
4.50/1.94	   POL(U52(x_1)) = x_1
4.50/1.94	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
4.50/1.94	   POL(U62(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
4.50/1.94	   POL(isNat(x_1)) = 0
4.50/1.94	   POL(isNatIList(x_1)) = 1 + x_1
4.50/1.94	   POL(isNatList(x_1)) = 0
4.50/1.94	   POL(length(x_1)) = x_1
4.50/1.94	   POL(s(x_1)) = x_1
4.50/1.94	   POL(tt) = 0
4.50/1.94	   POL(zeros) = 1
4.50/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.94	
4.50/1.94	   isNatIList(V) -> U31(isNatList(V))
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(10)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(11) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	Used ordering:
4.50/1.94	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(zeros) =  	[[0], [1]]
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(0) =  	[[0], [0]]
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U11(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(tt) =  	[[0], [1]]
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U21(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(isNatIList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U61(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(U62(x_1, x_2)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	   <<<
4.50/1.94	 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
4.50/1.94	>>>
4.50/1.94	
4.50/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.94	
4.50/1.94	   U62(tt, L) -> s(length(L))
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(12)
4.50/1.94	Obligation:
4.50/1.94	Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	
4.50/1.94	----------------------------------------
4.50/1.94	
4.50/1.94	(13) CSRRRRProof (EQUIVALENT)
4.50/1.94	The following CSR is given: Context-sensitive rewrite system:
4.50/1.94	The TRS R consists of the following rules:
4.50/1.94	
4.50/1.94	   zeros -> cons(0, zeros)
4.50/1.94	   U11(tt) -> tt
4.50/1.94	   U21(tt) -> tt
4.50/1.94	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.94	   U42(tt) -> tt
4.50/1.94	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.94	   U52(tt) -> tt
4.50/1.94	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.94	   isNat(0) -> tt
4.50/1.94	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.94	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.94	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.94	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.94	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.94	
4.50/1.94	The replacement map contains the following entries:
4.50/1.94	
4.50/1.94	zeros: empty set
4.50/1.94	cons: {1}
4.50/1.94	0: empty set
4.50/1.94	U11: {1}
4.50/1.94	tt: empty set
4.50/1.94	U21: {1}
4.50/1.94	U41: {1}
4.50/1.94	U42: {1}
4.50/1.94	isNatIList: empty set
4.50/1.94	U51: {1}
4.50/1.94	U52: {1}
4.50/1.94	isNatList: empty set
4.50/1.94	U61: {1}
4.50/1.94	U62: {1}
4.50/1.94	isNat: empty set
4.50/1.94	s: {1}
4.50/1.94	length: {1}
4.50/1.94	Used ordering:
4.50/1.94	Polynomial interpretation [POLO]:
4.50/1.94	
4.50/1.94	   POL(0) = 0
4.50/1.94	   POL(U11(x_1)) = x_1
4.50/1.94	   POL(U21(x_1)) = x_1
4.50/1.94	   POL(U41(x_1, x_2)) = x_1
4.50/1.94	   POL(U42(x_1)) = x_1
4.50/1.94	   POL(U51(x_1, x_2)) = x_1
4.50/1.94	   POL(U52(x_1)) = x_1
4.50/1.95	   POL(U61(x_1, x_2, x_3)) = 1 + x_1
4.50/1.95	   POL(U62(x_1, x_2)) = x_1
4.50/1.95	   POL(cons(x_1, x_2)) = 1 + x_1
4.50/1.95	   POL(isNat(x_1)) = 0
4.50/1.95	   POL(isNatIList(x_1)) = 0
4.50/1.95	   POL(isNatList(x_1)) = 0
4.50/1.95	   POL(length(x_1)) = 1 + x_1
4.50/1.95	   POL(s(x_1)) = 1 + x_1
4.50/1.95	   POL(tt) = 0
4.50/1.95	   POL(zeros) = 1
4.50/1.95	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.95	
4.50/1.95	   U61(tt, L, N) -> U62(isNat(N), L)
4.50/1.95	   length(cons(N, L)) -> U61(isNatList(L), L, N)
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(14)
4.50/1.95	Obligation:
4.50/1.95	Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   zeros -> cons(0, zeros)
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   U21(tt) -> tt
4.50/1.95	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.95	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	zeros: empty set
4.50/1.95	cons: {1}
4.50/1.95	0: empty set
4.50/1.95	U11: {1}
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U41: {1}
4.50/1.95	U42: {1}
4.50/1.95	isNatIList: empty set
4.50/1.95	U51: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNatList: empty set
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	length: {1}
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(15) CSRRRRProof (EQUIVALENT)
4.50/1.95	The following CSR is given: Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   zeros -> cons(0, zeros)
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   U21(tt) -> tt
4.50/1.95	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.95	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	zeros: empty set
4.50/1.95	cons: {1}
4.50/1.95	0: empty set
4.50/1.95	U11: {1}
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U41: {1}
4.50/1.95	U42: {1}
4.50/1.95	isNatIList: empty set
4.50/1.95	U51: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNatList: empty set
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	length: {1}
4.50/1.95	Used ordering:
4.50/1.95	Polynomial interpretation [POLO]:
4.50/1.95	
4.50/1.95	   POL(0) = 0
4.50/1.95	   POL(U11(x_1)) = x_1
4.50/1.95	   POL(U21(x_1)) = x_1
4.50/1.95	   POL(U41(x_1, x_2)) = x_1
4.50/1.95	   POL(U42(x_1)) = x_1
4.50/1.95	   POL(U51(x_1, x_2)) = x_1
4.50/1.95	   POL(U52(x_1)) = x_1
4.50/1.95	   POL(cons(x_1, x_2)) = x_1
4.50/1.95	   POL(isNat(x_1)) = 0
4.50/1.95	   POL(isNatIList(x_1)) = 0
4.50/1.95	   POL(isNatList(x_1)) = 0
4.50/1.95	   POL(length(x_1)) = 1 + x_1
4.50/1.95	   POL(s(x_1)) = 1 + x_1
4.50/1.95	   POL(tt) = 0
4.50/1.95	   POL(zeros) = 1
4.50/1.95	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.95	
4.50/1.95	   zeros -> cons(0, zeros)
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(16)
4.50/1.95	Obligation:
4.50/1.95	Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   U21(tt) -> tt
4.50/1.95	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.95	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	cons: {1}
4.50/1.95	0: empty set
4.50/1.95	U11: {1}
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U41: {1}
4.50/1.95	U42: {1}
4.50/1.95	isNatIList: empty set
4.50/1.95	U51: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNatList: empty set
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	length: {1}
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(17) CSRRRRProof (EQUIVALENT)
4.50/1.95	The following CSR is given: Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   U21(tt) -> tt
4.50/1.95	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.95	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	cons: {1}
4.50/1.95	0: empty set
4.50/1.95	U11: {1}
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U41: {1}
4.50/1.95	U42: {1}
4.50/1.95	isNatIList: empty set
4.50/1.95	U51: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNatList: empty set
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	length: {1}
4.50/1.95	Used ordering:
4.50/1.95	Polynomial interpretation [POLO]:
4.50/1.95	
4.50/1.95	   POL(0) = 1
4.50/1.95	   POL(U11(x_1)) = x_1
4.50/1.95	   POL(U21(x_1)) = 1 + x_1
4.50/1.95	   POL(U41(x_1, x_2)) = x_1 + x_2
4.50/1.95	   POL(U42(x_1)) = x_1
4.50/1.95	   POL(U51(x_1, x_2)) = x_1 + x_2
4.50/1.95	   POL(U52(x_1)) = x_1
4.50/1.95	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
4.50/1.95	   POL(isNat(x_1)) = x_1
4.50/1.95	   POL(isNatIList(x_1)) = x_1
4.50/1.95	   POL(isNatList(x_1)) = x_1
4.50/1.95	   POL(length(x_1)) = 1 + x_1
4.50/1.95	   POL(s(x_1)) = 1 + x_1
4.50/1.95	   POL(tt) = 1
4.50/1.95	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.95	
4.50/1.95	   U21(tt) -> tt
4.50/1.95	   U41(tt, V2) -> U42(isNatIList(V2))
4.50/1.95	   U51(tt, V2) -> U52(isNatList(V2))
4.50/1.95	   isNat(length(V1)) -> U11(isNatList(V1))
4.50/1.95	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
4.50/1.95	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(18)
4.50/1.95	Obligation:
4.50/1.95	Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	0: empty set
4.50/1.95	U11: {1}
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U42: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(19) CSRRRRProof (EQUIVALENT)
4.50/1.95	The following CSR is given: Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	0: empty set
4.50/1.95	U11: {1}
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U42: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	Used ordering:
4.50/1.95	Polynomial interpretation [POLO]:
4.50/1.95	
4.50/1.95	   POL(0) = 1
4.50/1.95	   POL(U11(x_1)) = 1 + x_1
4.50/1.95	   POL(U21(x_1)) = 1 + x_1
4.50/1.95	   POL(U42(x_1)) = x_1
4.50/1.95	   POL(U52(x_1)) = x_1
4.50/1.95	   POL(isNat(x_1)) = 1 + x_1
4.50/1.95	   POL(s(x_1)) = 1 + x_1
4.50/1.95	   POL(tt) = 0
4.50/1.95	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.95	
4.50/1.95	   U11(tt) -> tt
4.50/1.95	   isNat(0) -> tt
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(20)
4.50/1.95	Obligation:
4.50/1.95	Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U42: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(21) CSRRRRProof (EQUIVALENT)
4.50/1.95	The following CSR is given: Context-sensitive rewrite system:
4.50/1.95	The TRS R consists of the following rules:
4.50/1.95	
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	
4.50/1.95	The replacement map contains the following entries:
4.50/1.95	
4.50/1.95	tt: empty set
4.50/1.95	U21: {1}
4.50/1.95	U42: {1}
4.50/1.95	U52: {1}
4.50/1.95	isNat: empty set
4.50/1.95	s: {1}
4.50/1.95	Used ordering:
4.50/1.95	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.50/1.95	
4.50/1.95	   <<<
4.50/1.95	 POL(U42(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.50/1.95	>>>
4.50/1.95	
4.50/1.95	   <<<
4.50/1.95	 POL(tt) =  	[[1], [1]]
4.50/1.95	>>>
4.50/1.95	
4.50/1.95	   <<<
4.50/1.95	 POL(U52(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.50/1.95	>>>
4.50/1.95	
4.50/1.95	   <<<
4.50/1.95	 POL(isNat(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.50/1.95	>>>
4.50/1.95	
4.50/1.95	   <<<
4.50/1.95	 POL(s(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.50/1.95	>>>
4.50/1.95	
4.50/1.95	   <<<
4.50/1.95	 POL(U21(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
4.50/1.95	>>>
4.50/1.95	
4.50/1.95	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.50/1.95	
4.50/1.95	   U42(tt) -> tt
4.50/1.95	   U52(tt) -> tt
4.50/1.95	   isNat(s(V1)) -> U21(isNat(V1))
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(22)
4.50/1.95	Obligation:
4.50/1.95	Context-sensitive rewrite system:
4.50/1.95	R is empty.
4.50/1.95	
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(23) RisEmptyProof (EQUIVALENT)
4.50/1.95	The CSR R is empty. Hence, termination is trivially proven.
4.50/1.95	----------------------------------------
4.50/1.95	
4.50/1.95	(24)
4.50/1.95	YES
4.50/1.98	EOF