5.51/2.80	YES
5.51/2.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.51/2.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.51/2.81	
5.51/2.81	
5.51/2.81	Termination w.r.t. Q of the given QTRS could be proven:
5.51/2.81	
5.51/2.81	(0) QTRS
5.51/2.81	(1) QTRSToCSRProof [SOUND, 0 ms]
5.51/2.81	(2) CSR
5.51/2.81	(3) CSRRRRProof [EQUIVALENT, 87 ms]
5.51/2.81	(4) CSR
5.51/2.81	(5) CSRRRRProof [EQUIVALENT, 0 ms]
5.51/2.81	(6) CSR
5.51/2.81	(7) CSRRRRProof [EQUIVALENT, 10 ms]
5.51/2.81	(8) CSR
5.51/2.81	(9) CSRRRRProof [EQUIVALENT, 18 ms]
5.51/2.81	(10) CSR
5.51/2.81	(11) CSRRRRProof [EQUIVALENT, 0 ms]
5.51/2.81	(12) CSR
5.51/2.81	(13) CSRRRRProof [EQUIVALENT, 9 ms]
5.51/2.81	(14) CSR
5.51/2.81	(15) CSRRRRProof [EQUIVALENT, 0 ms]
5.51/2.81	(16) CSR
5.51/2.81	(17) CSRRRRProof [EQUIVALENT, 0 ms]
5.51/2.81	(18) CSR
5.51/2.81	(19) CSRRRRProof [EQUIVALENT, 0 ms]
5.51/2.81	(20) CSR
5.51/2.81	(21) CSRRRRProof [EQUIVALENT, 0 ms]
5.51/2.81	(22) CSR
5.51/2.81	(23) RisEmptyProof [EQUIVALENT, 0 ms]
5.51/2.81	(24) YES
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(0)
5.51/2.81	Obligation:
5.51/2.81	Q restricted rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   active(zeros) -> mark(cons(0, zeros))
5.51/2.81	   active(U11(tt)) -> mark(tt)
5.51/2.81	   active(U21(tt)) -> mark(tt)
5.51/2.81	   active(U31(tt)) -> mark(tt)
5.51/2.81	   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
5.51/2.81	   active(U42(tt)) -> mark(tt)
5.51/2.81	   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
5.51/2.81	   active(U52(tt)) -> mark(tt)
5.51/2.81	   active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
5.51/2.81	   active(U62(tt, L)) -> mark(s(length(L)))
5.51/2.81	   active(isNat(0)) -> mark(tt)
5.51/2.81	   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
5.51/2.81	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
5.51/2.81	   active(isNatIList(V)) -> mark(U31(isNatList(V)))
5.51/2.81	   active(isNatIList(zeros)) -> mark(tt)
5.51/2.81	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
5.51/2.81	   active(isNatList(nil)) -> mark(tt)
5.51/2.81	   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
5.51/2.81	   active(length(nil)) -> mark(0)
5.51/2.81	   active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
5.51/2.81	   active(cons(X1, X2)) -> cons(active(X1), X2)
5.51/2.81	   active(U11(X)) -> U11(active(X))
5.51/2.81	   active(U21(X)) -> U21(active(X))
5.51/2.81	   active(U31(X)) -> U31(active(X))
5.51/2.81	   active(U41(X1, X2)) -> U41(active(X1), X2)
5.51/2.81	   active(U42(X)) -> U42(active(X))
5.51/2.81	   active(U51(X1, X2)) -> U51(active(X1), X2)
5.51/2.81	   active(U52(X)) -> U52(active(X))
5.51/2.81	   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
5.51/2.81	   active(U62(X1, X2)) -> U62(active(X1), X2)
5.51/2.81	   active(s(X)) -> s(active(X))
5.51/2.81	   active(length(X)) -> length(active(X))
5.51/2.81	   cons(mark(X1), X2) -> mark(cons(X1, X2))
5.51/2.81	   U11(mark(X)) -> mark(U11(X))
5.51/2.81	   U21(mark(X)) -> mark(U21(X))
5.51/2.81	   U31(mark(X)) -> mark(U31(X))
5.51/2.81	   U41(mark(X1), X2) -> mark(U41(X1, X2))
5.51/2.81	   U42(mark(X)) -> mark(U42(X))
5.51/2.81	   U51(mark(X1), X2) -> mark(U51(X1, X2))
5.51/2.81	   U52(mark(X)) -> mark(U52(X))
5.51/2.81	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
5.51/2.81	   U62(mark(X1), X2) -> mark(U62(X1, X2))
5.51/2.81	   s(mark(X)) -> mark(s(X))
5.51/2.81	   length(mark(X)) -> mark(length(X))
5.51/2.81	   proper(zeros) -> ok(zeros)
5.51/2.81	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
5.51/2.81	   proper(0) -> ok(0)
5.51/2.81	   proper(U11(X)) -> U11(proper(X))
5.51/2.81	   proper(tt) -> ok(tt)
5.51/2.81	   proper(U21(X)) -> U21(proper(X))
5.51/2.81	   proper(U31(X)) -> U31(proper(X))
5.51/2.81	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
5.51/2.81	   proper(U42(X)) -> U42(proper(X))
5.51/2.81	   proper(isNatIList(X)) -> isNatIList(proper(X))
5.51/2.81	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
5.51/2.81	   proper(U52(X)) -> U52(proper(X))
5.51/2.81	   proper(isNatList(X)) -> isNatList(proper(X))
5.51/2.81	   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
5.51/2.81	   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
5.51/2.81	   proper(isNat(X)) -> isNat(proper(X))
5.51/2.81	   proper(s(X)) -> s(proper(X))
5.51/2.81	   proper(length(X)) -> length(proper(X))
5.51/2.81	   proper(nil) -> ok(nil)
5.51/2.81	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
5.51/2.81	   U11(ok(X)) -> ok(U11(X))
5.51/2.81	   U21(ok(X)) -> ok(U21(X))
5.51/2.81	   U31(ok(X)) -> ok(U31(X))
5.51/2.81	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
5.51/2.81	   U42(ok(X)) -> ok(U42(X))
5.51/2.81	   isNatIList(ok(X)) -> ok(isNatIList(X))
5.51/2.81	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
5.51/2.81	   U52(ok(X)) -> ok(U52(X))
5.51/2.81	   isNatList(ok(X)) -> ok(isNatList(X))
5.51/2.81	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
5.51/2.81	   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
5.51/2.81	   isNat(ok(X)) -> ok(isNat(X))
5.51/2.81	   s(ok(X)) -> ok(s(X))
5.51/2.81	   length(ok(X)) -> ok(length(X))
5.51/2.81	   top(mark(X)) -> top(proper(X))
5.51/2.81	   top(ok(X)) -> top(active(X))
5.51/2.81	
5.51/2.81	The set Q consists of the following terms:
5.51/2.81	
5.51/2.81	   active(zeros)
5.51/2.81	   active(isNat(0))
5.51/2.81	   active(isNat(length(x0)))
5.51/2.81	   active(isNat(s(x0)))
5.51/2.81	   active(isNatIList(x0))
5.51/2.81	   active(isNatList(nil))
5.51/2.81	   active(isNatList(cons(x0, x1)))
5.51/2.81	   active(cons(x0, x1))
5.51/2.81	   active(U11(x0))
5.51/2.81	   active(U21(x0))
5.51/2.81	   active(U31(x0))
5.51/2.81	   active(U41(x0, x1))
5.51/2.81	   active(U42(x0))
5.51/2.81	   active(U51(x0, x1))
5.51/2.81	   active(U52(x0))
5.51/2.81	   active(U61(x0, x1, x2))
5.51/2.81	   active(U62(x0, x1))
5.51/2.81	   active(s(x0))
5.51/2.81	   active(length(x0))
5.51/2.81	   cons(mark(x0), x1)
5.51/2.81	   U11(mark(x0))
5.51/2.81	   U21(mark(x0))
5.51/2.81	   U31(mark(x0))
5.51/2.81	   U41(mark(x0), x1)
5.51/2.81	   U42(mark(x0))
5.51/2.81	   U51(mark(x0), x1)
5.51/2.81	   U52(mark(x0))
5.51/2.81	   U61(mark(x0), x1, x2)
5.51/2.81	   U62(mark(x0), x1)
5.51/2.81	   s(mark(x0))
5.51/2.81	   length(mark(x0))
5.51/2.81	   proper(zeros)
5.51/2.81	   proper(cons(x0, x1))
5.51/2.81	   proper(0)
5.51/2.81	   proper(U11(x0))
5.51/2.81	   proper(tt)
5.51/2.81	   proper(U21(x0))
5.51/2.81	   proper(U31(x0))
5.51/2.81	   proper(U41(x0, x1))
5.51/2.81	   proper(U42(x0))
5.51/2.81	   proper(isNatIList(x0))
5.51/2.81	   proper(U51(x0, x1))
5.51/2.81	   proper(U52(x0))
5.51/2.81	   proper(isNatList(x0))
5.51/2.81	   proper(U61(x0, x1, x2))
5.51/2.81	   proper(U62(x0, x1))
5.51/2.81	   proper(isNat(x0))
5.51/2.81	   proper(s(x0))
5.51/2.81	   proper(length(x0))
5.51/2.81	   proper(nil)
5.51/2.81	   cons(ok(x0), ok(x1))
5.51/2.81	   U11(ok(x0))
5.51/2.81	   U21(ok(x0))
5.51/2.81	   U31(ok(x0))
5.51/2.81	   U41(ok(x0), ok(x1))
5.51/2.81	   U42(ok(x0))
5.51/2.81	   isNatIList(ok(x0))
5.51/2.81	   U51(ok(x0), ok(x1))
5.51/2.81	   U52(ok(x0))
5.51/2.81	   isNatList(ok(x0))
5.51/2.81	   U61(ok(x0), ok(x1), ok(x2))
5.51/2.81	   U62(ok(x0), ok(x1))
5.51/2.81	   isNat(ok(x0))
5.51/2.81	   s(ok(x0))
5.51/2.81	   length(ok(x0))
5.51/2.81	   top(mark(x0))
5.51/2.81	   top(ok(x0))
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(1) QTRSToCSRProof (SOUND)
5.51/2.81	The following Q TRS is given: Q restricted rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   active(zeros) -> mark(cons(0, zeros))
5.51/2.81	   active(U11(tt)) -> mark(tt)
5.51/2.81	   active(U21(tt)) -> mark(tt)
5.51/2.81	   active(U31(tt)) -> mark(tt)
5.51/2.81	   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
5.51/2.81	   active(U42(tt)) -> mark(tt)
5.51/2.81	   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
5.51/2.81	   active(U52(tt)) -> mark(tt)
5.51/2.81	   active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
5.51/2.81	   active(U62(tt, L)) -> mark(s(length(L)))
5.51/2.81	   active(isNat(0)) -> mark(tt)
5.51/2.81	   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
5.51/2.81	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
5.51/2.81	   active(isNatIList(V)) -> mark(U31(isNatList(V)))
5.51/2.81	   active(isNatIList(zeros)) -> mark(tt)
5.51/2.81	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
5.51/2.81	   active(isNatList(nil)) -> mark(tt)
5.51/2.81	   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
5.51/2.81	   active(length(nil)) -> mark(0)
5.51/2.81	   active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
5.51/2.81	   active(cons(X1, X2)) -> cons(active(X1), X2)
5.51/2.81	   active(U11(X)) -> U11(active(X))
5.51/2.81	   active(U21(X)) -> U21(active(X))
5.51/2.81	   active(U31(X)) -> U31(active(X))
5.51/2.81	   active(U41(X1, X2)) -> U41(active(X1), X2)
5.51/2.81	   active(U42(X)) -> U42(active(X))
5.51/2.81	   active(U51(X1, X2)) -> U51(active(X1), X2)
5.51/2.81	   active(U52(X)) -> U52(active(X))
5.51/2.81	   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
5.51/2.81	   active(U62(X1, X2)) -> U62(active(X1), X2)
5.51/2.81	   active(s(X)) -> s(active(X))
5.51/2.81	   active(length(X)) -> length(active(X))
5.51/2.81	   cons(mark(X1), X2) -> mark(cons(X1, X2))
5.51/2.81	   U11(mark(X)) -> mark(U11(X))
5.51/2.81	   U21(mark(X)) -> mark(U21(X))
5.51/2.81	   U31(mark(X)) -> mark(U31(X))
5.51/2.81	   U41(mark(X1), X2) -> mark(U41(X1, X2))
5.51/2.81	   U42(mark(X)) -> mark(U42(X))
5.51/2.81	   U51(mark(X1), X2) -> mark(U51(X1, X2))
5.51/2.81	   U52(mark(X)) -> mark(U52(X))
5.51/2.81	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
5.51/2.81	   U62(mark(X1), X2) -> mark(U62(X1, X2))
5.51/2.81	   s(mark(X)) -> mark(s(X))
5.51/2.81	   length(mark(X)) -> mark(length(X))
5.51/2.81	   proper(zeros) -> ok(zeros)
5.51/2.81	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
5.51/2.81	   proper(0) -> ok(0)
5.51/2.81	   proper(U11(X)) -> U11(proper(X))
5.51/2.81	   proper(tt) -> ok(tt)
5.51/2.81	   proper(U21(X)) -> U21(proper(X))
5.51/2.81	   proper(U31(X)) -> U31(proper(X))
5.51/2.81	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
5.51/2.81	   proper(U42(X)) -> U42(proper(X))
5.51/2.81	   proper(isNatIList(X)) -> isNatIList(proper(X))
5.51/2.81	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
5.51/2.81	   proper(U52(X)) -> U52(proper(X))
5.51/2.81	   proper(isNatList(X)) -> isNatList(proper(X))
5.51/2.81	   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
5.51/2.81	   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
5.51/2.81	   proper(isNat(X)) -> isNat(proper(X))
5.51/2.81	   proper(s(X)) -> s(proper(X))
5.51/2.81	   proper(length(X)) -> length(proper(X))
5.51/2.81	   proper(nil) -> ok(nil)
5.51/2.81	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
5.51/2.81	   U11(ok(X)) -> ok(U11(X))
5.51/2.81	   U21(ok(X)) -> ok(U21(X))
5.51/2.81	   U31(ok(X)) -> ok(U31(X))
5.51/2.81	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
5.51/2.81	   U42(ok(X)) -> ok(U42(X))
5.51/2.81	   isNatIList(ok(X)) -> ok(isNatIList(X))
5.51/2.81	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
5.51/2.81	   U52(ok(X)) -> ok(U52(X))
5.51/2.81	   isNatList(ok(X)) -> ok(isNatList(X))
5.51/2.81	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
5.51/2.81	   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
5.51/2.81	   isNat(ok(X)) -> ok(isNat(X))
5.51/2.81	   s(ok(X)) -> ok(s(X))
5.51/2.81	   length(ok(X)) -> ok(length(X))
5.51/2.81	   top(mark(X)) -> top(proper(X))
5.51/2.81	   top(ok(X)) -> top(active(X))
5.51/2.81	
5.51/2.81	The set Q consists of the following terms:
5.51/2.81	
5.51/2.81	   active(zeros)
5.51/2.81	   active(isNat(0))
5.51/2.81	   active(isNat(length(x0)))
5.51/2.81	   active(isNat(s(x0)))
5.51/2.81	   active(isNatIList(x0))
5.51/2.81	   active(isNatList(nil))
5.51/2.81	   active(isNatList(cons(x0, x1)))
5.51/2.81	   active(cons(x0, x1))
5.51/2.81	   active(U11(x0))
5.51/2.81	   active(U21(x0))
5.51/2.81	   active(U31(x0))
5.51/2.81	   active(U41(x0, x1))
5.51/2.81	   active(U42(x0))
5.51/2.81	   active(U51(x0, x1))
5.51/2.81	   active(U52(x0))
5.51/2.81	   active(U61(x0, x1, x2))
5.51/2.81	   active(U62(x0, x1))
5.51/2.81	   active(s(x0))
5.51/2.81	   active(length(x0))
5.51/2.81	   cons(mark(x0), x1)
5.51/2.81	   U11(mark(x0))
5.51/2.81	   U21(mark(x0))
5.51/2.81	   U31(mark(x0))
5.51/2.81	   U41(mark(x0), x1)
5.51/2.81	   U42(mark(x0))
5.51/2.81	   U51(mark(x0), x1)
5.51/2.81	   U52(mark(x0))
5.51/2.81	   U61(mark(x0), x1, x2)
5.51/2.81	   U62(mark(x0), x1)
5.51/2.81	   s(mark(x0))
5.51/2.81	   length(mark(x0))
5.51/2.81	   proper(zeros)
5.51/2.81	   proper(cons(x0, x1))
5.51/2.81	   proper(0)
5.51/2.81	   proper(U11(x0))
5.51/2.81	   proper(tt)
5.51/2.81	   proper(U21(x0))
5.51/2.81	   proper(U31(x0))
5.51/2.81	   proper(U41(x0, x1))
5.51/2.81	   proper(U42(x0))
5.51/2.81	   proper(isNatIList(x0))
5.51/2.81	   proper(U51(x0, x1))
5.51/2.81	   proper(U52(x0))
5.51/2.81	   proper(isNatList(x0))
5.51/2.81	   proper(U61(x0, x1, x2))
5.51/2.81	   proper(U62(x0, x1))
5.51/2.81	   proper(isNat(x0))
5.51/2.81	   proper(s(x0))
5.51/2.81	   proper(length(x0))
5.51/2.81	   proper(nil)
5.51/2.81	   cons(ok(x0), ok(x1))
5.51/2.81	   U11(ok(x0))
5.51/2.81	   U21(ok(x0))
5.51/2.81	   U31(ok(x0))
5.51/2.81	   U41(ok(x0), ok(x1))
5.51/2.81	   U42(ok(x0))
5.51/2.81	   isNatIList(ok(x0))
5.51/2.81	   U51(ok(x0), ok(x1))
5.51/2.81	   U52(ok(x0))
5.51/2.81	   isNatList(ok(x0))
5.51/2.81	   U61(ok(x0), ok(x1), ok(x2))
5.51/2.81	   U62(ok(x0), ok(x1))
5.51/2.81	   isNat(ok(x0))
5.51/2.81	   s(ok(x0))
5.51/2.81	   length(ok(x0))
5.51/2.81	   top(mark(x0))
5.51/2.81	   top(ok(x0))
5.51/2.81	
5.51/2.81	Special symbols used for the transformation (see [GM04]):
5.51/2.81	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(2)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U31(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(V) -> U31(isNatList(V))
5.51/2.81	   isNatIList(zeros) -> tt
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(nil) -> 0
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(3) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U31(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(V) -> U31(isNatList(V))
5.51/2.81	   isNatIList(zeros) -> tt
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(nil) -> 0
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = 2*x_1
5.51/2.81	   POL(U21(x_1)) = 2*x_1
5.51/2.81	   POL(U31(x_1)) = 2*x_1
5.51/2.81	   POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2
5.51/2.81	   POL(U42(x_1)) = x_1
5.51/2.81	   POL(U51(x_1, x_2)) = 2*x_1
5.51/2.81	   POL(U52(x_1)) = x_1
5.51/2.81	   POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
5.51/2.81	   POL(U62(x_1, x_2)) = 2*x_1 + 2*x_2
5.51/2.81	   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
5.51/2.81	   POL(isNat(x_1)) = 0
5.51/2.81	   POL(isNatIList(x_1)) = 1 + x_1
5.51/2.81	   POL(isNatList(x_1)) = 0
5.51/2.81	   POL(length(x_1)) = 2*x_1
5.51/2.81	   POL(nil) = 0
5.51/2.81	   POL(s(x_1)) = x_1
5.51/2.81	   POL(tt) = 0
5.51/2.81	   POL(zeros) = 0
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   isNatIList(V) -> U31(isNatList(V))
5.51/2.81	   isNatIList(zeros) -> tt
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(4)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U31(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(nil) -> 0
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(5) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U31(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(nil) -> 0
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = x_1
5.51/2.81	   POL(U21(x_1)) = x_1
5.51/2.81	   POL(U31(x_1)) = x_1
5.51/2.81	   POL(U41(x_1, x_2)) = x_1 + x_2
5.51/2.81	   POL(U42(x_1)) = x_1
5.51/2.81	   POL(U51(x_1, x_2)) = x_1
5.51/2.81	   POL(U52(x_1)) = x_1
5.51/2.81	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
5.51/2.81	   POL(U62(x_1, x_2)) = x_1 + x_2
5.51/2.81	   POL(cons(x_1, x_2)) = x_1 + x_2
5.51/2.81	   POL(isNat(x_1)) = 1
5.51/2.81	   POL(isNatIList(x_1)) = 1 + x_1
5.51/2.81	   POL(isNatList(x_1)) = 1
5.51/2.81	   POL(length(x_1)) = 1 + x_1
5.51/2.81	   POL(nil) = 1
5.51/2.81	   POL(s(x_1)) = x_1
5.51/2.81	   POL(tt) = 1
5.51/2.81	   POL(zeros) = 1
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   length(nil) -> 0
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(6)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U31(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(7) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U31(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U31: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = x_1
5.51/2.81	   POL(U21(x_1)) = x_1
5.51/2.81	   POL(U31(x_1)) = 1 + x_1
5.51/2.81	   POL(U41(x_1, x_2)) = x_1 + x_2
5.51/2.81	   POL(U42(x_1)) = x_1
5.51/2.81	   POL(U51(x_1, x_2)) = x_1
5.51/2.81	   POL(U52(x_1)) = x_1
5.51/2.81	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
5.51/2.81	   POL(U62(x_1, x_2)) = x_1 + x_2
5.51/2.81	   POL(cons(x_1, x_2)) = x_1 + x_2
5.51/2.81	   POL(isNat(x_1)) = 1
5.51/2.81	   POL(isNatIList(x_1)) = 1 + x_1
5.51/2.81	   POL(isNatList(x_1)) = 1
5.51/2.81	   POL(length(x_1)) = 1 + x_1
5.51/2.81	   POL(nil) = 1
5.51/2.81	   POL(s(x_1)) = x_1
5.51/2.81	   POL(tt) = 1
5.51/2.81	   POL(zeros) = 1
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   U31(tt) -> tt
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(8)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(9) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	nil: empty set
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = x_1
5.51/2.81	   POL(U21(x_1)) = x_1
5.51/2.81	   POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2
5.51/2.81	   POL(U42(x_1)) = x_1
5.51/2.81	   POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2
5.51/2.81	   POL(U52(x_1)) = 2*x_1
5.51/2.81	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
5.51/2.81	   POL(U62(x_1, x_2)) = 2*x_1 + x_2
5.51/2.81	   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
5.51/2.81	   POL(isNat(x_1)) = x_1
5.51/2.81	   POL(isNatIList(x_1)) = 2*x_1
5.51/2.81	   POL(isNatList(x_1)) = x_1
5.51/2.81	   POL(length(x_1)) = x_1
5.51/2.81	   POL(nil) = 1
5.51/2.81	   POL(s(x_1)) = x_1
5.51/2.81	   POL(tt) = 0
5.51/2.81	   POL(zeros) = 0
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   isNatList(nil) -> tt
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(10)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(11) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	Used ordering:
5.51/2.81	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(zeros) =  	[[0], [1]]
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(0) =  	[[0], [0]]
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U11(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(tt) =  	[[0], [1]]
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U21(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(isNatIList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 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
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(U62(x_1, x_2)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	   <<<
5.51/2.81	 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
5.51/2.81	>>>
5.51/2.81	
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   U62(tt, L) -> s(length(L))
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(12)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(13) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = 2*x_1
5.51/2.81	   POL(U21(x_1)) = x_1
5.51/2.81	   POL(U41(x_1, x_2)) = x_1
5.51/2.81	   POL(U42(x_1)) = 2*x_1
5.51/2.81	   POL(U51(x_1, x_2)) = 2*x_1
5.51/2.81	   POL(U52(x_1)) = x_1
5.51/2.81	   POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_3
5.51/2.81	   POL(U62(x_1, x_2)) = x_1
5.51/2.81	   POL(cons(x_1, x_2)) = 2 + 2*x_1
5.51/2.81	   POL(isNat(x_1)) = 0
5.51/2.81	   POL(isNatIList(x_1)) = 0
5.51/2.81	   POL(isNatList(x_1)) = 0
5.51/2.81	   POL(length(x_1)) = 1 + 2*x_1
5.51/2.81	   POL(s(x_1)) = 2 + 2*x_1
5.51/2.81	   POL(tt) = 0
5.51/2.81	   POL(zeros) = 2
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   length(cons(N, L)) -> U61(isNatList(L), L, N)
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(14)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(15) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	U61: {1}
5.51/2.81	U62: {1}
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = x_1
5.51/2.81	   POL(U21(x_1)) = x_1
5.51/2.81	   POL(U41(x_1, x_2)) = 2*x_1
5.51/2.81	   POL(U42(x_1)) = 2*x_1
5.51/2.81	   POL(U51(x_1, x_2)) = 2*x_1
5.51/2.81	   POL(U52(x_1)) = 2*x_1
5.51/2.81	   POL(U61(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
5.51/2.81	   POL(U62(x_1, x_2)) = 2*x_1 + 2*x_2
5.51/2.81	   POL(cons(x_1, x_2)) = x_1
5.51/2.81	   POL(isNat(x_1)) = 0
5.51/2.81	   POL(isNatIList(x_1)) = 0
5.51/2.81	   POL(isNatList(x_1)) = 0
5.51/2.81	   POL(length(x_1)) = 2 + 2*x_1
5.51/2.81	   POL(s(x_1)) = 2 + 2*x_1
5.51/2.81	   POL(tt) = 0
5.51/2.81	   POL(zeros) = 0
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.81	
5.51/2.81	   U61(tt, L, N) -> U62(isNat(N), L)
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(16)
5.51/2.81	Obligation:
5.51/2.81	Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	
5.51/2.81	----------------------------------------
5.51/2.81	
5.51/2.81	(17) CSRRRRProof (EQUIVALENT)
5.51/2.81	The following CSR is given: Context-sensitive rewrite system:
5.51/2.81	The TRS R consists of the following rules:
5.51/2.81	
5.51/2.81	   zeros -> cons(0, zeros)
5.51/2.81	   U11(tt) -> tt
5.51/2.81	   U21(tt) -> tt
5.51/2.81	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.81	   U42(tt) -> tt
5.51/2.81	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.81	   U52(tt) -> tt
5.51/2.81	   isNat(0) -> tt
5.51/2.81	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.81	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.81	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.81	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.81	
5.51/2.81	The replacement map contains the following entries:
5.51/2.81	
5.51/2.81	zeros: empty set
5.51/2.81	cons: {1}
5.51/2.81	0: empty set
5.51/2.81	U11: {1}
5.51/2.81	tt: empty set
5.51/2.81	U21: {1}
5.51/2.81	U41: {1}
5.51/2.81	U42: {1}
5.51/2.81	isNatIList: empty set
5.51/2.81	U51: {1}
5.51/2.81	U52: {1}
5.51/2.81	isNatList: empty set
5.51/2.81	isNat: empty set
5.51/2.81	s: {1}
5.51/2.81	length: {1}
5.51/2.81	Used ordering:
5.51/2.81	Polynomial interpretation [POLO]:
5.51/2.81	
5.51/2.81	   POL(0) = 0
5.51/2.81	   POL(U11(x_1)) = x_1
5.51/2.81	   POL(U21(x_1)) = x_1
5.51/2.81	   POL(U41(x_1, x_2)) = x_1
5.51/2.81	   POL(U42(x_1)) = x_1
5.51/2.81	   POL(U51(x_1, x_2)) = x_1
5.51/2.81	   POL(U52(x_1)) = x_1
5.51/2.81	   POL(cons(x_1, x_2)) = x_1
5.51/2.81	   POL(isNat(x_1)) = 0
5.51/2.81	   POL(isNatIList(x_1)) = 0
5.51/2.81	   POL(isNatList(x_1)) = 0
5.51/2.81	   POL(length(x_1)) = 1 + x_1
5.51/2.81	   POL(s(x_1)) = 1 + x_1
5.51/2.81	   POL(tt) = 0
5.51/2.81	   POL(zeros) = 1
5.51/2.81	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.82	
5.51/2.82	   zeros -> cons(0, zeros)
5.51/2.82	
5.51/2.82	
5.51/2.82	
5.51/2.82	
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(18)
5.51/2.82	Obligation:
5.51/2.82	Context-sensitive rewrite system:
5.51/2.82	The TRS R consists of the following rules:
5.51/2.82	
5.51/2.82	   U11(tt) -> tt
5.51/2.82	   U21(tt) -> tt
5.51/2.82	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.82	   U42(tt) -> tt
5.51/2.82	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.82	   U52(tt) -> tt
5.51/2.82	   isNat(0) -> tt
5.51/2.82	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.82	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.82	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.82	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.82	
5.51/2.82	The replacement map contains the following entries:
5.51/2.82	
5.51/2.82	cons: {1}
5.51/2.82	0: empty set
5.51/2.82	U11: {1}
5.51/2.82	tt: empty set
5.51/2.82	U21: {1}
5.51/2.82	U41: {1}
5.51/2.82	U42: {1}
5.51/2.82	isNatIList: empty set
5.51/2.82	U51: {1}
5.51/2.82	U52: {1}
5.51/2.82	isNatList: empty set
5.51/2.82	isNat: empty set
5.51/2.82	s: {1}
5.51/2.82	length: {1}
5.51/2.82	
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(19) CSRRRRProof (EQUIVALENT)
5.51/2.82	The following CSR is given: Context-sensitive rewrite system:
5.51/2.82	The TRS R consists of the following rules:
5.51/2.82	
5.51/2.82	   U11(tt) -> tt
5.51/2.82	   U21(tt) -> tt
5.51/2.82	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.82	   U42(tt) -> tt
5.51/2.82	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.82	   U52(tt) -> tt
5.51/2.82	   isNat(0) -> tt
5.51/2.82	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.82	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.82	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.82	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.82	
5.51/2.82	The replacement map contains the following entries:
5.51/2.82	
5.51/2.82	cons: {1}
5.51/2.82	0: empty set
5.51/2.82	U11: {1}
5.51/2.82	tt: empty set
5.51/2.82	U21: {1}
5.51/2.82	U41: {1}
5.51/2.82	U42: {1}
5.51/2.82	isNatIList: empty set
5.51/2.82	U51: {1}
5.51/2.82	U52: {1}
5.51/2.82	isNatList: empty set
5.51/2.82	isNat: empty set
5.51/2.82	s: {1}
5.51/2.82	length: {1}
5.51/2.82	Used ordering:
5.51/2.82	Polynomial interpretation [POLO]:
5.51/2.82	
5.51/2.82	   POL(0) = 1
5.51/2.82	   POL(U11(x_1)) = x_1
5.51/2.82	   POL(U21(x_1)) = 1 + x_1
5.51/2.82	   POL(U41(x_1, x_2)) = x_1 + x_2
5.51/2.82	   POL(U42(x_1)) = x_1
5.51/2.82	   POL(U51(x_1, x_2)) = x_1 + x_2
5.51/2.82	   POL(U52(x_1)) = x_1
5.51/2.82	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
5.51/2.82	   POL(isNat(x_1)) = x_1
5.51/2.82	   POL(isNatIList(x_1)) = x_1
5.51/2.82	   POL(isNatList(x_1)) = x_1
5.51/2.82	   POL(length(x_1)) = 1 + x_1
5.51/2.82	   POL(s(x_1)) = 1 + x_1
5.51/2.82	   POL(tt) = 1
5.51/2.82	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.82	
5.51/2.82	   U21(tt) -> tt
5.51/2.82	   U41(tt, V2) -> U42(isNatIList(V2))
5.51/2.82	   U51(tt, V2) -> U52(isNatList(V2))
5.51/2.82	   isNat(length(V1)) -> U11(isNatList(V1))
5.51/2.82	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
5.51/2.82	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
5.51/2.82	
5.51/2.82	
5.51/2.82	
5.51/2.82	
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(20)
5.51/2.82	Obligation:
5.51/2.82	Context-sensitive rewrite system:
5.51/2.82	The TRS R consists of the following rules:
5.51/2.82	
5.51/2.82	   U11(tt) -> tt
5.51/2.82	   U42(tt) -> tt
5.51/2.82	   U52(tt) -> tt
5.51/2.82	   isNat(0) -> tt
5.51/2.82	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.82	
5.51/2.82	The replacement map contains the following entries:
5.51/2.82	
5.51/2.82	0: empty set
5.51/2.82	U11: {1}
5.51/2.82	tt: empty set
5.51/2.82	U21: {1}
5.51/2.82	U42: {1}
5.51/2.82	U52: {1}
5.51/2.82	isNat: empty set
5.51/2.82	s: {1}
5.51/2.82	
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(21) CSRRRRProof (EQUIVALENT)
5.51/2.82	The following CSR is given: Context-sensitive rewrite system:
5.51/2.82	The TRS R consists of the following rules:
5.51/2.82	
5.51/2.82	   U11(tt) -> tt
5.51/2.82	   U42(tt) -> tt
5.51/2.82	   U52(tt) -> tt
5.51/2.82	   isNat(0) -> tt
5.51/2.82	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.82	
5.51/2.82	The replacement map contains the following entries:
5.51/2.82	
5.51/2.82	0: empty set
5.51/2.82	U11: {1}
5.51/2.82	tt: empty set
5.51/2.82	U21: {1}
5.51/2.82	U42: {1}
5.51/2.82	U52: {1}
5.51/2.82	isNat: empty set
5.51/2.82	s: {1}
5.51/2.82	Used ordering:
5.51/2.82	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(U11(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(tt) =  	[[1], [1]]
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(U42(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(U52(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(isNat(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(0) =  	[[1], [1]]
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(s(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	   <<<
5.51/2.82	 POL(U21(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
5.51/2.82	>>>
5.51/2.82	
5.51/2.82	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.51/2.82	
5.51/2.82	   U11(tt) -> tt
5.51/2.82	   U42(tt) -> tt
5.51/2.82	   U52(tt) -> tt
5.51/2.82	   isNat(0) -> tt
5.51/2.82	   isNat(s(V1)) -> U21(isNat(V1))
5.51/2.82	
5.51/2.82	
5.51/2.82	
5.51/2.82	
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(22)
5.51/2.82	Obligation:
5.51/2.82	Context-sensitive rewrite system:
5.51/2.82	R is empty.
5.51/2.82	
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(23) RisEmptyProof (EQUIVALENT)
5.51/2.82	The CSR R is empty. Hence, termination is trivially proven.
5.51/2.82	----------------------------------------
5.51/2.82	
5.51/2.82	(24)
5.51/2.82	YES
5.51/2.84	EOF