7.19/2.75	YES
7.43/2.77	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
7.43/2.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.43/2.77	
7.43/2.77	
7.43/2.77	Termination w.r.t. Q of the given QTRS could be proven:
7.43/2.77	
7.43/2.77	(0) QTRS
7.43/2.77	(1) QTRSToCSRProof [SOUND, 0 ms]
7.43/2.77	(2) CSR
7.43/2.77	(3) CSRRRRProof [EQUIVALENT, 87 ms]
7.43/2.77	(4) CSR
7.43/2.77	(5) CSRRRRProof [EQUIVALENT, 20 ms]
7.43/2.77	(6) CSR
7.43/2.77	(7) CSRRRRProof [EQUIVALENT, 15 ms]
7.43/2.77	(8) CSR
7.43/2.77	(9) CSRRRRProof [EQUIVALENT, 23 ms]
7.43/2.77	(10) CSR
7.43/2.77	(11) CSRRRRProof [EQUIVALENT, 13 ms]
7.43/2.77	(12) CSR
7.43/2.77	(13) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(14) CSR
7.43/2.77	(15) CSRRRRProof [EQUIVALENT, 10 ms]
7.43/2.77	(16) CSR
7.43/2.77	(17) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(18) CSR
7.43/2.77	(19) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(20) CSR
7.43/2.77	(21) CSRRRRProof [EQUIVALENT, 7 ms]
7.43/2.77	(22) CSR
7.43/2.77	(23) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(24) CSR
7.43/2.77	(25) CSRRRRProof [EQUIVALENT, 6 ms]
7.43/2.77	(26) CSR
7.43/2.77	(27) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(28) CSR
7.43/2.77	(29) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(30) CSR
7.43/2.77	(31) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(32) CSR
7.43/2.77	(33) CSRRRRProof [EQUIVALENT, 0 ms]
7.43/2.77	(34) CSR
7.43/2.77	(35) CSRRRRProof [EQUIVALENT, 2 ms]
7.43/2.77	(36) CSR
7.43/2.77	(37) RisEmptyProof [EQUIVALENT, 0 ms]
7.43/2.77	(38) YES
7.43/2.77	
7.43/2.77	
7.43/2.77	----------------------------------------
7.43/2.77	
7.43/2.77	(0)
7.43/2.77	Obligation:
7.43/2.77	Q restricted rewrite system:
7.43/2.77	The TRS R consists of the following rules:
7.43/2.77	
7.43/2.77	   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
7.43/2.77	   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
7.43/2.77	   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
7.43/2.77	   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
7.43/2.77	   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
7.43/2.77	   active(U16(tt)) -> mark(tt)
7.43/2.77	   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
7.43/2.77	   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
7.43/2.77	   active(U23(tt)) -> mark(tt)
7.43/2.77	   active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
7.43/2.77	   active(U32(tt)) -> mark(tt)
7.43/2.77	   active(U41(tt)) -> mark(tt)
7.43/2.77	   active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
7.43/2.77	   active(U52(tt, N)) -> mark(N)
7.43/2.77	   active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
7.43/2.77	   active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
7.43/2.77	   active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
7.43/2.77	   active(U64(tt, M, N)) -> mark(s(plus(N, M)))
7.43/2.77	   active(isNat(0)) -> mark(tt)
7.43/2.77	   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
7.43/2.77	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
7.43/2.77	   active(isNatKind(0)) -> mark(tt)
7.43/2.77	   active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
7.43/2.77	   active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
7.43/2.77	   active(plus(N, 0)) -> mark(U51(isNat(N), N))
7.43/2.77	   active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
7.43/2.77	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
7.43/2.77	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
7.43/2.77	   active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
7.43/2.77	   active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
7.43/2.77	   active(U15(X1, X2)) -> U15(active(X1), X2)
7.43/2.77	   active(U16(X)) -> U16(active(X))
7.43/2.77	   active(U21(X1, X2)) -> U21(active(X1), X2)
7.43/2.77	   active(U22(X1, X2)) -> U22(active(X1), X2)
7.43/2.77	   active(U23(X)) -> U23(active(X))
7.43/2.77	   active(U31(X1, X2)) -> U31(active(X1), X2)
7.43/2.77	   active(U32(X)) -> U32(active(X))
7.43/2.77	   active(U41(X)) -> U41(active(X))
7.43/2.77	   active(U51(X1, X2)) -> U51(active(X1), X2)
7.43/2.77	   active(U52(X1, X2)) -> U52(active(X1), X2)
7.43/2.77	   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
7.43/2.77	   active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
7.43/2.77	   active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
7.43/2.77	   active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
7.43/2.77	   active(s(X)) -> s(active(X))
7.43/2.77	   active(plus(X1, X2)) -> plus(active(X1), X2)
7.43/2.77	   active(plus(X1, X2)) -> plus(X1, active(X2))
7.43/2.77	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
7.43/2.77	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
7.43/2.77	   U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
7.43/2.77	   U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
7.43/2.77	   U15(mark(X1), X2) -> mark(U15(X1, X2))
7.43/2.77	   U16(mark(X)) -> mark(U16(X))
7.43/2.77	   U21(mark(X1), X2) -> mark(U21(X1, X2))
7.43/2.77	   U22(mark(X1), X2) -> mark(U22(X1, X2))
7.43/2.77	   U23(mark(X)) -> mark(U23(X))
7.43/2.77	   U31(mark(X1), X2) -> mark(U31(X1, X2))
7.43/2.77	   U32(mark(X)) -> mark(U32(X))
7.43/2.77	   U41(mark(X)) -> mark(U41(X))
7.43/2.77	   U51(mark(X1), X2) -> mark(U51(X1, X2))
7.43/2.77	   U52(mark(X1), X2) -> mark(U52(X1, X2))
7.43/2.77	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
7.43/2.77	   U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
7.43/2.77	   U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
7.43/2.77	   U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
7.43/2.77	   s(mark(X)) -> mark(s(X))
7.43/2.77	   plus(mark(X1), X2) -> mark(plus(X1, X2))
7.43/2.77	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
7.43/2.77	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(tt) -> ok(tt)
7.43/2.77	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(isNatKind(X)) -> isNatKind(proper(X))
7.43/2.77	   proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
7.43/2.77	   proper(isNat(X)) -> isNat(proper(X))
7.43/2.77	   proper(U16(X)) -> U16(proper(X))
7.43/2.77	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
7.43/2.77	   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
7.43/2.77	   proper(U23(X)) -> U23(proper(X))
7.43/2.77	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
7.43/2.77	   proper(U32(X)) -> U32(proper(X))
7.43/2.77	   proper(U41(X)) -> U41(proper(X))
7.43/2.77	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
7.43/2.77	   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
7.43/2.77	   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(s(X)) -> s(proper(X))
7.43/2.77	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
7.43/2.77	   proper(0) -> ok(0)
7.43/2.77	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
7.43/2.77	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
7.43/2.77	   isNatKind(ok(X)) -> ok(isNatKind(X))
7.43/2.77	   U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
7.43/2.77	   U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
7.43/2.77	   U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
7.43/2.77	   isNat(ok(X)) -> ok(isNat(X))
7.43/2.77	   U16(ok(X)) -> ok(U16(X))
7.43/2.77	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
7.43/2.77	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
7.43/2.77	   U23(ok(X)) -> ok(U23(X))
7.43/2.77	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
7.43/2.77	   U32(ok(X)) -> ok(U32(X))
7.43/2.77	   U41(ok(X)) -> ok(U41(X))
7.43/2.77	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
7.43/2.77	   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
7.43/2.77	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
7.43/2.77	   U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
7.43/2.77	   U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
7.43/2.77	   U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
7.43/2.77	   s(ok(X)) -> ok(s(X))
7.43/2.77	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
7.43/2.77	   top(mark(X)) -> top(proper(X))
7.43/2.77	   top(ok(X)) -> top(active(X))
7.43/2.77	
7.43/2.77	The set Q consists of the following terms:
7.43/2.77	
7.43/2.77	   active(isNat(0))
7.43/2.77	   active(isNat(plus(x0, x1)))
7.43/2.77	   active(isNat(s(x0)))
7.43/2.77	   active(isNatKind(0))
7.43/2.77	   active(isNatKind(plus(x0, x1)))
7.43/2.77	   active(isNatKind(s(x0)))
7.43/2.77	   active(U11(x0, x1, x2))
7.43/2.77	   active(U12(x0, x1, x2))
7.43/2.77	   active(U13(x0, x1, x2))
7.43/2.77	   active(U14(x0, x1, x2))
7.43/2.77	   active(U15(x0, x1))
7.43/2.77	   active(U16(x0))
7.43/2.77	   active(U21(x0, x1))
7.43/2.77	   active(U22(x0, x1))
7.43/2.77	   active(U23(x0))
7.43/2.77	   active(U31(x0, x1))
7.43/2.77	   active(U32(x0))
7.43/2.77	   active(U41(x0))
7.43/2.77	   active(U51(x0, x1))
7.43/2.77	   active(U52(x0, x1))
7.43/2.77	   active(U61(x0, x1, x2))
7.43/2.77	   active(U62(x0, x1, x2))
7.43/2.77	   active(U63(x0, x1, x2))
7.43/2.77	   active(U64(x0, x1, x2))
7.43/2.77	   active(s(x0))
7.43/2.77	   active(plus(x0, x1))
7.43/2.77	   U11(mark(x0), x1, x2)
7.43/2.77	   U12(mark(x0), x1, x2)
7.43/2.77	   U13(mark(x0), x1, x2)
7.43/2.77	   U14(mark(x0), x1, x2)
7.43/2.77	   U15(mark(x0), x1)
7.43/2.77	   U16(mark(x0))
7.43/2.77	   U21(mark(x0), x1)
7.43/2.77	   U22(mark(x0), x1)
7.43/2.77	   U23(mark(x0))
7.43/2.77	   U31(mark(x0), x1)
7.43/2.77	   U32(mark(x0))
7.43/2.77	   U41(mark(x0))
7.43/2.77	   U51(mark(x0), x1)
7.43/2.77	   U52(mark(x0), x1)
7.43/2.77	   U61(mark(x0), x1, x2)
7.43/2.77	   U62(mark(x0), x1, x2)
7.43/2.77	   U63(mark(x0), x1, x2)
7.43/2.77	   U64(mark(x0), x1, x2)
7.43/2.77	   s(mark(x0))
7.43/2.77	   plus(mark(x0), x1)
7.43/2.77	   plus(x0, mark(x1))
7.43/2.77	   proper(U11(x0, x1, x2))
7.43/2.77	   proper(tt)
7.43/2.77	   proper(U12(x0, x1, x2))
7.43/2.77	   proper(isNatKind(x0))
7.43/2.77	   proper(U13(x0, x1, x2))
7.43/2.77	   proper(U14(x0, x1, x2))
7.43/2.77	   proper(U15(x0, x1))
7.43/2.77	   proper(isNat(x0))
7.43/2.77	   proper(U16(x0))
7.43/2.77	   proper(U21(x0, x1))
7.43/2.77	   proper(U22(x0, x1))
7.43/2.77	   proper(U23(x0))
7.43/2.77	   proper(U31(x0, x1))
7.43/2.77	   proper(U32(x0))
7.43/2.77	   proper(U41(x0))
7.43/2.77	   proper(U51(x0, x1))
7.43/2.77	   proper(U52(x0, x1))
7.43/2.77	   proper(U61(x0, x1, x2))
7.43/2.77	   proper(U62(x0, x1, x2))
7.43/2.77	   proper(U63(x0, x1, x2))
7.43/2.77	   proper(U64(x0, x1, x2))
7.43/2.77	   proper(s(x0))
7.43/2.77	   proper(plus(x0, x1))
7.43/2.77	   proper(0)
7.43/2.77	   U11(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U12(ok(x0), ok(x1), ok(x2))
7.43/2.77	   isNatKind(ok(x0))
7.43/2.77	   U13(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U14(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U15(ok(x0), ok(x1))
7.43/2.77	   isNat(ok(x0))
7.43/2.77	   U16(ok(x0))
7.43/2.77	   U21(ok(x0), ok(x1))
7.43/2.77	   U22(ok(x0), ok(x1))
7.43/2.77	   U23(ok(x0))
7.43/2.77	   U31(ok(x0), ok(x1))
7.43/2.77	   U32(ok(x0))
7.43/2.77	   U41(ok(x0))
7.43/2.77	   U51(ok(x0), ok(x1))
7.43/2.77	   U52(ok(x0), ok(x1))
7.43/2.77	   U61(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U62(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U63(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U64(ok(x0), ok(x1), ok(x2))
7.43/2.77	   s(ok(x0))
7.43/2.77	   plus(ok(x0), ok(x1))
7.43/2.77	   top(mark(x0))
7.43/2.77	   top(ok(x0))
7.43/2.77	
7.43/2.77	
7.43/2.77	----------------------------------------
7.43/2.77	
7.43/2.77	(1) QTRSToCSRProof (SOUND)
7.43/2.77	The following Q TRS is given: Q restricted rewrite system:
7.43/2.77	The TRS R consists of the following rules:
7.43/2.77	
7.43/2.77	   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
7.43/2.77	   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
7.43/2.77	   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
7.43/2.77	   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
7.43/2.77	   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
7.43/2.77	   active(U16(tt)) -> mark(tt)
7.43/2.77	   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
7.43/2.77	   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
7.43/2.77	   active(U23(tt)) -> mark(tt)
7.43/2.77	   active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
7.43/2.77	   active(U32(tt)) -> mark(tt)
7.43/2.77	   active(U41(tt)) -> mark(tt)
7.43/2.77	   active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
7.43/2.77	   active(U52(tt, N)) -> mark(N)
7.43/2.77	   active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
7.43/2.77	   active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
7.43/2.77	   active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
7.43/2.77	   active(U64(tt, M, N)) -> mark(s(plus(N, M)))
7.43/2.77	   active(isNat(0)) -> mark(tt)
7.43/2.77	   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
7.43/2.77	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
7.43/2.77	   active(isNatKind(0)) -> mark(tt)
7.43/2.77	   active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
7.43/2.77	   active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
7.43/2.77	   active(plus(N, 0)) -> mark(U51(isNat(N), N))
7.43/2.77	   active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
7.43/2.77	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
7.43/2.77	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
7.43/2.77	   active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
7.43/2.77	   active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
7.43/2.77	   active(U15(X1, X2)) -> U15(active(X1), X2)
7.43/2.77	   active(U16(X)) -> U16(active(X))
7.43/2.77	   active(U21(X1, X2)) -> U21(active(X1), X2)
7.43/2.77	   active(U22(X1, X2)) -> U22(active(X1), X2)
7.43/2.77	   active(U23(X)) -> U23(active(X))
7.43/2.77	   active(U31(X1, X2)) -> U31(active(X1), X2)
7.43/2.77	   active(U32(X)) -> U32(active(X))
7.43/2.77	   active(U41(X)) -> U41(active(X))
7.43/2.77	   active(U51(X1, X2)) -> U51(active(X1), X2)
7.43/2.77	   active(U52(X1, X2)) -> U52(active(X1), X2)
7.43/2.77	   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
7.43/2.77	   active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
7.43/2.77	   active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
7.43/2.77	   active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
7.43/2.77	   active(s(X)) -> s(active(X))
7.43/2.77	   active(plus(X1, X2)) -> plus(active(X1), X2)
7.43/2.77	   active(plus(X1, X2)) -> plus(X1, active(X2))
7.43/2.77	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
7.43/2.77	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
7.43/2.77	   U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
7.43/2.77	   U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
7.43/2.77	   U15(mark(X1), X2) -> mark(U15(X1, X2))
7.43/2.77	   U16(mark(X)) -> mark(U16(X))
7.43/2.77	   U21(mark(X1), X2) -> mark(U21(X1, X2))
7.43/2.77	   U22(mark(X1), X2) -> mark(U22(X1, X2))
7.43/2.77	   U23(mark(X)) -> mark(U23(X))
7.43/2.77	   U31(mark(X1), X2) -> mark(U31(X1, X2))
7.43/2.77	   U32(mark(X)) -> mark(U32(X))
7.43/2.77	   U41(mark(X)) -> mark(U41(X))
7.43/2.77	   U51(mark(X1), X2) -> mark(U51(X1, X2))
7.43/2.77	   U52(mark(X1), X2) -> mark(U52(X1, X2))
7.43/2.77	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
7.43/2.77	   U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
7.43/2.77	   U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
7.43/2.77	   U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
7.43/2.77	   s(mark(X)) -> mark(s(X))
7.43/2.77	   plus(mark(X1), X2) -> mark(plus(X1, X2))
7.43/2.77	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
7.43/2.77	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(tt) -> ok(tt)
7.43/2.77	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(isNatKind(X)) -> isNatKind(proper(X))
7.43/2.77	   proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
7.43/2.77	   proper(isNat(X)) -> isNat(proper(X))
7.43/2.77	   proper(U16(X)) -> U16(proper(X))
7.43/2.77	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
7.43/2.77	   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
7.43/2.77	   proper(U23(X)) -> U23(proper(X))
7.43/2.77	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
7.43/2.77	   proper(U32(X)) -> U32(proper(X))
7.43/2.77	   proper(U41(X)) -> U41(proper(X))
7.43/2.77	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
7.43/2.77	   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
7.43/2.77	   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
7.43/2.77	   proper(s(X)) -> s(proper(X))
7.43/2.77	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
7.43/2.77	   proper(0) -> ok(0)
7.43/2.77	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
7.43/2.77	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
7.43/2.77	   isNatKind(ok(X)) -> ok(isNatKind(X))
7.43/2.77	   U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
7.43/2.77	   U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
7.43/2.77	   U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
7.43/2.77	   isNat(ok(X)) -> ok(isNat(X))
7.43/2.77	   U16(ok(X)) -> ok(U16(X))
7.43/2.77	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
7.43/2.77	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
7.43/2.77	   U23(ok(X)) -> ok(U23(X))
7.43/2.77	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
7.43/2.77	   U32(ok(X)) -> ok(U32(X))
7.43/2.77	   U41(ok(X)) -> ok(U41(X))
7.43/2.77	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
7.43/2.77	   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
7.43/2.77	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
7.43/2.77	   U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
7.43/2.77	   U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
7.43/2.77	   U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
7.43/2.77	   s(ok(X)) -> ok(s(X))
7.43/2.77	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
7.43/2.77	   top(mark(X)) -> top(proper(X))
7.43/2.77	   top(ok(X)) -> top(active(X))
7.43/2.77	
7.43/2.77	The set Q consists of the following terms:
7.43/2.77	
7.43/2.77	   active(isNat(0))
7.43/2.77	   active(isNat(plus(x0, x1)))
7.43/2.77	   active(isNat(s(x0)))
7.43/2.77	   active(isNatKind(0))
7.43/2.77	   active(isNatKind(plus(x0, x1)))
7.43/2.77	   active(isNatKind(s(x0)))
7.43/2.77	   active(U11(x0, x1, x2))
7.43/2.77	   active(U12(x0, x1, x2))
7.43/2.77	   active(U13(x0, x1, x2))
7.43/2.77	   active(U14(x0, x1, x2))
7.43/2.77	   active(U15(x0, x1))
7.43/2.77	   active(U16(x0))
7.43/2.77	   active(U21(x0, x1))
7.43/2.77	   active(U22(x0, x1))
7.43/2.77	   active(U23(x0))
7.43/2.77	   active(U31(x0, x1))
7.43/2.77	   active(U32(x0))
7.43/2.77	   active(U41(x0))
7.43/2.77	   active(U51(x0, x1))
7.43/2.77	   active(U52(x0, x1))
7.43/2.77	   active(U61(x0, x1, x2))
7.43/2.77	   active(U62(x0, x1, x2))
7.43/2.77	   active(U63(x0, x1, x2))
7.43/2.77	   active(U64(x0, x1, x2))
7.43/2.77	   active(s(x0))
7.43/2.77	   active(plus(x0, x1))
7.43/2.77	   U11(mark(x0), x1, x2)
7.43/2.77	   U12(mark(x0), x1, x2)
7.43/2.77	   U13(mark(x0), x1, x2)
7.43/2.77	   U14(mark(x0), x1, x2)
7.43/2.77	   U15(mark(x0), x1)
7.43/2.77	   U16(mark(x0))
7.43/2.77	   U21(mark(x0), x1)
7.43/2.77	   U22(mark(x0), x1)
7.43/2.77	   U23(mark(x0))
7.43/2.77	   U31(mark(x0), x1)
7.43/2.77	   U32(mark(x0))
7.43/2.77	   U41(mark(x0))
7.43/2.77	   U51(mark(x0), x1)
7.43/2.77	   U52(mark(x0), x1)
7.43/2.77	   U61(mark(x0), x1, x2)
7.43/2.77	   U62(mark(x0), x1, x2)
7.43/2.77	   U63(mark(x0), x1, x2)
7.43/2.77	   U64(mark(x0), x1, x2)
7.43/2.77	   s(mark(x0))
7.43/2.77	   plus(mark(x0), x1)
7.43/2.77	   plus(x0, mark(x1))
7.43/2.77	   proper(U11(x0, x1, x2))
7.43/2.77	   proper(tt)
7.43/2.77	   proper(U12(x0, x1, x2))
7.43/2.77	   proper(isNatKind(x0))
7.43/2.77	   proper(U13(x0, x1, x2))
7.43/2.77	   proper(U14(x0, x1, x2))
7.43/2.77	   proper(U15(x0, x1))
7.43/2.77	   proper(isNat(x0))
7.43/2.77	   proper(U16(x0))
7.43/2.77	   proper(U21(x0, x1))
7.43/2.77	   proper(U22(x0, x1))
7.43/2.77	   proper(U23(x0))
7.43/2.77	   proper(U31(x0, x1))
7.43/2.77	   proper(U32(x0))
7.43/2.77	   proper(U41(x0))
7.43/2.77	   proper(U51(x0, x1))
7.43/2.77	   proper(U52(x0, x1))
7.43/2.77	   proper(U61(x0, x1, x2))
7.43/2.77	   proper(U62(x0, x1, x2))
7.43/2.77	   proper(U63(x0, x1, x2))
7.43/2.77	   proper(U64(x0, x1, x2))
7.43/2.77	   proper(s(x0))
7.43/2.77	   proper(plus(x0, x1))
7.43/2.77	   proper(0)
7.43/2.77	   U11(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U12(ok(x0), ok(x1), ok(x2))
7.43/2.77	   isNatKind(ok(x0))
7.43/2.77	   U13(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U14(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U15(ok(x0), ok(x1))
7.43/2.77	   isNat(ok(x0))
7.43/2.77	   U16(ok(x0))
7.43/2.77	   U21(ok(x0), ok(x1))
7.43/2.77	   U22(ok(x0), ok(x1))
7.43/2.77	   U23(ok(x0))
7.43/2.77	   U31(ok(x0), ok(x1))
7.43/2.77	   U32(ok(x0))
7.43/2.77	   U41(ok(x0))
7.43/2.77	   U51(ok(x0), ok(x1))
7.43/2.77	   U52(ok(x0), ok(x1))
7.43/2.77	   U61(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U62(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U63(ok(x0), ok(x1), ok(x2))
7.43/2.77	   U64(ok(x0), ok(x1), ok(x2))
7.43/2.77	   s(ok(x0))
7.43/2.77	   plus(ok(x0), ok(x1))
7.43/2.77	   top(mark(x0))
7.43/2.77	   top(ok(x0))
7.43/2.77	
7.43/2.77	Special symbols used for the transformation (see [GM04]):
7.43/2.77	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
7.43/2.77	The replacement map contains the following entries:
7.43/2.77	
7.43/2.77	U11: {1}
7.43/2.77	tt: empty set
7.43/2.77	U12: {1}
7.43/2.77	isNatKind: empty set
7.43/2.77	U13: {1}
7.43/2.77	U14: {1}
7.43/2.77	U15: {1}
7.43/2.77	isNat: empty set
7.43/2.77	U16: {1}
7.43/2.77	U21: {1}
7.43/2.77	U22: {1}
7.43/2.77	U23: {1}
7.43/2.77	U31: {1}
7.43/2.77	U32: {1}
7.43/2.77	U41: {1}
7.43/2.77	U51: {1}
7.43/2.77	U52: {1}
7.43/2.77	U61: {1}
7.43/2.77	U62: {1}
7.43/2.77	U63: {1}
7.43/2.77	U64: {1}
7.43/2.77	s: {1}
7.43/2.77	plus: {1, 2}
7.43/2.77	0: empty set
7.43/2.77	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.
7.43/2.77	----------------------------------------
7.43/2.77	
7.43/2.77	(2)
7.43/2.77	Obligation:
7.43/2.77	Context-sensitive rewrite system:
7.43/2.77	The TRS R consists of the following rules:
7.43/2.77	
7.43/2.77	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.77	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.77	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.77	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.77	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.77	   U16(tt) -> tt
7.43/2.77	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.77	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.77	   U23(tt) -> tt
7.43/2.77	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.77	   U32(tt) -> tt
7.43/2.77	   U41(tt) -> tt
7.43/2.77	   U51(tt, N) -> U52(isNatKind(N), N)
7.43/2.77	   U52(tt, N) -> N
7.43/2.77	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.77	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.77	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.77	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.77	   isNat(0) -> tt
7.43/2.77	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.77	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.77	   isNatKind(0) -> tt
7.43/2.77	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.77	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.77	   plus(N, 0) -> U51(isNat(N), N)
7.43/2.77	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.77	
7.43/2.77	The replacement map contains the following entries:
7.43/2.77	
7.43/2.77	U11: {1}
7.43/2.77	tt: empty set
7.43/2.77	U12: {1}
7.43/2.77	isNatKind: empty set
7.43/2.77	U13: {1}
7.43/2.77	U14: {1}
7.43/2.77	U15: {1}
7.43/2.77	isNat: empty set
7.43/2.77	U16: {1}
7.43/2.77	U21: {1}
7.43/2.77	U22: {1}
7.43/2.77	U23: {1}
7.43/2.77	U31: {1}
7.43/2.77	U32: {1}
7.43/2.77	U41: {1}
7.43/2.77	U51: {1}
7.43/2.77	U52: {1}
7.43/2.77	U61: {1}
7.43/2.77	U62: {1}
7.43/2.77	U63: {1}
7.43/2.77	U64: {1}
7.43/2.77	s: {1}
7.43/2.77	plus: {1, 2}
7.43/2.77	0: empty set
7.43/2.77	
7.43/2.77	----------------------------------------
7.43/2.77	
7.43/2.77	(3) CSRRRRProof (EQUIVALENT)
7.43/2.77	The following CSR is given: Context-sensitive rewrite system:
7.43/2.77	The TRS R consists of the following rules:
7.43/2.77	
7.43/2.77	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.77	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.77	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.77	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.77	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.77	   U16(tt) -> tt
7.43/2.77	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.77	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.77	   U23(tt) -> tt
7.43/2.77	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.77	   U32(tt) -> tt
7.43/2.77	   U41(tt) -> tt
7.43/2.77	   U51(tt, N) -> U52(isNatKind(N), N)
7.43/2.77	   U52(tt, N) -> N
7.43/2.77	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.77	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.77	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.77	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.77	   isNat(0) -> tt
7.43/2.77	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.77	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.77	   isNatKind(0) -> tt
7.43/2.77	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.77	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.77	   plus(N, 0) -> U51(isNat(N), N)
7.43/2.77	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.77	
7.43/2.77	The replacement map contains the following entries:
7.43/2.77	
7.43/2.77	U11: {1}
7.43/2.77	tt: empty set
7.43/2.77	U12: {1}
7.43/2.77	isNatKind: empty set
7.43/2.77	U13: {1}
7.43/2.77	U14: {1}
7.43/2.77	U15: {1}
7.43/2.77	isNat: empty set
7.43/2.77	U16: {1}
7.43/2.77	U21: {1}
7.43/2.77	U22: {1}
7.43/2.77	U23: {1}
7.43/2.77	U31: {1}
7.43/2.77	U32: {1}
7.43/2.77	U41: {1}
7.43/2.77	U51: {1}
7.43/2.77	U52: {1}
7.43/2.77	U61: {1}
7.43/2.77	U62: {1}
7.43/2.77	U63: {1}
7.43/2.77	U64: {1}
7.43/2.77	s: {1}
7.43/2.77	plus: {1, 2}
7.43/2.77	0: empty set
7.43/2.77	Used ordering:
7.43/2.77	Polynomial interpretation [POLO]:
7.43/2.77	
7.43/2.77	   POL(0) = 1
7.43/2.77	   POL(U11(x_1, x_2, x_3)) = x_1
7.43/2.77	   POL(U12(x_1, x_2, x_3)) = x_1
7.43/2.77	   POL(U13(x_1, x_2, x_3)) = x_1
7.43/2.77	   POL(U14(x_1, x_2, x_3)) = x_1
7.43/2.77	   POL(U15(x_1, x_2)) = x_1
7.43/2.77	   POL(U16(x_1)) = x_1
7.43/2.77	   POL(U21(x_1, x_2)) = x_1
7.43/2.77	   POL(U22(x_1, x_2)) = x_1
7.43/2.77	   POL(U23(x_1)) = x_1
7.43/2.77	   POL(U31(x_1, x_2)) = x_1
7.43/2.77	   POL(U32(x_1)) = x_1
7.43/2.77	   POL(U41(x_1)) = x_1
7.43/2.77	   POL(U51(x_1, x_2)) = x_1 + x_2
7.43/2.77	   POL(U52(x_1, x_2)) = x_1 + x_2
7.43/2.77	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.77	   POL(U62(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.77	   POL(U63(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.77	   POL(U64(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.77	   POL(isNat(x_1)) = 0
7.43/2.77	   POL(isNatKind(x_1)) = 0
7.43/2.77	   POL(plus(x_1, x_2)) = x_1 + x_2
7.43/2.77	   POL(s(x_1)) = x_1
7.43/2.77	   POL(tt) = 0
7.43/2.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.43/2.77	
7.43/2.77	   plus(N, 0) -> U51(isNat(N), N)
7.43/2.77	
7.43/2.77	
7.43/2.77	
7.43/2.77	
7.43/2.77	----------------------------------------
7.43/2.77	
7.43/2.77	(4)
7.43/2.77	Obligation:
7.43/2.77	Context-sensitive rewrite system:
7.43/2.77	The TRS R consists of the following rules:
7.43/2.77	
7.43/2.77	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.77	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.77	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.77	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.77	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.77	   U16(tt) -> tt
7.43/2.77	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.77	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.77	   U23(tt) -> tt
7.43/2.77	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.77	   U32(tt) -> tt
7.43/2.77	   U41(tt) -> tt
7.43/2.78	   U51(tt, N) -> U52(isNatKind(N), N)
7.43/2.78	   U52(tt, N) -> N
7.43/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.78	   isNat(0) -> tt
7.43/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.78	   isNatKind(0) -> tt
7.43/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.78	
7.43/2.78	The replacement map contains the following entries:
7.43/2.78	
7.43/2.78	U11: {1}
7.43/2.78	tt: empty set
7.43/2.78	U12: {1}
7.43/2.78	isNatKind: empty set
7.43/2.78	U13: {1}
7.43/2.78	U14: {1}
7.43/2.78	U15: {1}
7.43/2.78	isNat: empty set
7.43/2.78	U16: {1}
7.43/2.78	U21: {1}
7.43/2.78	U22: {1}
7.43/2.78	U23: {1}
7.43/2.78	U31: {1}
7.43/2.78	U32: {1}
7.43/2.78	U41: {1}
7.43/2.78	U51: {1}
7.43/2.78	U52: {1}
7.43/2.78	U61: {1}
7.43/2.78	U62: {1}
7.43/2.78	U63: {1}
7.43/2.78	U64: {1}
7.43/2.78	s: {1}
7.43/2.78	plus: {1, 2}
7.43/2.78	0: empty set
7.43/2.78	
7.43/2.78	----------------------------------------
7.43/2.78	
7.43/2.78	(5) CSRRRRProof (EQUIVALENT)
7.43/2.78	The following CSR is given: Context-sensitive rewrite system:
7.43/2.78	The TRS R consists of the following rules:
7.43/2.78	
7.43/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.78	   U16(tt) -> tt
7.43/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.78	   U23(tt) -> tt
7.43/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.78	   U32(tt) -> tt
7.43/2.78	   U41(tt) -> tt
7.43/2.78	   U51(tt, N) -> U52(isNatKind(N), N)
7.43/2.78	   U52(tt, N) -> N
7.43/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.78	   isNat(0) -> tt
7.43/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.78	   isNatKind(0) -> tt
7.43/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.78	
7.43/2.78	The replacement map contains the following entries:
7.43/2.78	
7.43/2.78	U11: {1}
7.43/2.78	tt: empty set
7.43/2.78	U12: {1}
7.43/2.78	isNatKind: empty set
7.43/2.78	U13: {1}
7.43/2.78	U14: {1}
7.43/2.78	U15: {1}
7.43/2.78	isNat: empty set
7.43/2.78	U16: {1}
7.43/2.78	U21: {1}
7.43/2.78	U22: {1}
7.43/2.78	U23: {1}
7.43/2.78	U31: {1}
7.43/2.78	U32: {1}
7.43/2.78	U41: {1}
7.43/2.78	U51: {1}
7.43/2.78	U52: {1}
7.43/2.78	U61: {1}
7.43/2.78	U62: {1}
7.43/2.78	U63: {1}
7.43/2.78	U64: {1}
7.43/2.78	s: {1}
7.43/2.78	plus: {1, 2}
7.43/2.78	0: empty set
7.43/2.78	Used ordering:
7.43/2.78	Polynomial interpretation [POLO]:
7.43/2.78	
7.43/2.78	   POL(0) = 1
7.43/2.78	   POL(U11(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U12(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U14(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U15(x_1, x_2)) = x_1
7.43/2.78	   POL(U16(x_1)) = x_1
7.43/2.78	   POL(U21(x_1, x_2)) = x_1
7.43/2.78	   POL(U22(x_1, x_2)) = x_1
7.43/2.78	   POL(U23(x_1)) = x_1
7.43/2.78	   POL(U31(x_1, x_2)) = x_1
7.43/2.78	   POL(U32(x_1)) = x_1
7.43/2.78	   POL(U41(x_1)) = x_1
7.43/2.78	   POL(U51(x_1, x_2)) = 1 + x_1 + x_2
7.43/2.78	   POL(U52(x_1, x_2)) = x_1 + x_2
7.43/2.78	   POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.78	   POL(U62(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.78	   POL(U63(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.78	   POL(U64(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.43/2.78	   POL(isNat(x_1)) = 0
7.43/2.78	   POL(isNatKind(x_1)) = 0
7.43/2.78	   POL(plus(x_1, x_2)) = x_1 + x_2
7.43/2.78	   POL(s(x_1)) = x_1
7.43/2.78	   POL(tt) = 0
7.43/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.43/2.78	
7.43/2.78	   U51(tt, N) -> U52(isNatKind(N), N)
7.43/2.78	
7.43/2.78	
7.43/2.78	
7.43/2.78	
7.43/2.78	----------------------------------------
7.43/2.78	
7.43/2.78	(6)
7.43/2.78	Obligation:
7.43/2.78	Context-sensitive rewrite system:
7.43/2.78	The TRS R consists of the following rules:
7.43/2.78	
7.43/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.78	   U16(tt) -> tt
7.43/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.78	   U23(tt) -> tt
7.43/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.78	   U32(tt) -> tt
7.43/2.78	   U41(tt) -> tt
7.43/2.78	   U52(tt, N) -> N
7.43/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.78	   isNat(0) -> tt
7.43/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.78	   isNatKind(0) -> tt
7.43/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.78	
7.43/2.78	The replacement map contains the following entries:
7.43/2.78	
7.43/2.78	U11: {1}
7.43/2.78	tt: empty set
7.43/2.78	U12: {1}
7.43/2.78	isNatKind: empty set
7.43/2.78	U13: {1}
7.43/2.78	U14: {1}
7.43/2.78	U15: {1}
7.43/2.78	isNat: empty set
7.43/2.78	U16: {1}
7.43/2.78	U21: {1}
7.43/2.78	U22: {1}
7.43/2.78	U23: {1}
7.43/2.78	U31: {1}
7.43/2.78	U32: {1}
7.43/2.78	U41: {1}
7.43/2.78	U52: {1}
7.43/2.78	U61: {1}
7.43/2.78	U62: {1}
7.43/2.78	U63: {1}
7.43/2.78	U64: {1}
7.43/2.78	s: {1}
7.43/2.78	plus: {1, 2}
7.43/2.78	0: empty set
7.43/2.78	
7.43/2.78	----------------------------------------
7.43/2.78	
7.43/2.78	(7) CSRRRRProof (EQUIVALENT)
7.43/2.78	The following CSR is given: Context-sensitive rewrite system:
7.43/2.78	The TRS R consists of the following rules:
7.43/2.78	
7.43/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.78	   U16(tt) -> tt
7.43/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.78	   U23(tt) -> tt
7.43/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.78	   U32(tt) -> tt
7.43/2.78	   U41(tt) -> tt
7.43/2.78	   U52(tt, N) -> N
7.43/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.78	   isNat(0) -> tt
7.43/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.78	   isNatKind(0) -> tt
7.43/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.78	
7.43/2.78	The replacement map contains the following entries:
7.43/2.78	
7.43/2.78	U11: {1}
7.43/2.78	tt: empty set
7.43/2.78	U12: {1}
7.43/2.78	isNatKind: empty set
7.43/2.78	U13: {1}
7.43/2.78	U14: {1}
7.43/2.78	U15: {1}
7.43/2.78	isNat: empty set
7.43/2.78	U16: {1}
7.43/2.78	U21: {1}
7.43/2.78	U22: {1}
7.43/2.78	U23: {1}
7.43/2.78	U31: {1}
7.43/2.78	U32: {1}
7.43/2.78	U41: {1}
7.43/2.78	U52: {1}
7.43/2.78	U61: {1}
7.43/2.78	U62: {1}
7.43/2.78	U63: {1}
7.43/2.78	U64: {1}
7.43/2.78	s: {1}
7.43/2.78	plus: {1, 2}
7.43/2.78	0: empty set
7.43/2.78	Used ordering:
7.43/2.78	Polynomial interpretation [POLO]:
7.43/2.78	
7.43/2.78	   POL(0) = 1
7.43/2.78	   POL(U11(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U12(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U14(x_1, x_2, x_3)) = x_1
7.43/2.78	   POL(U15(x_1, x_2)) = x_1
7.43/2.78	   POL(U16(x_1)) = x_1
7.43/2.78	   POL(U21(x_1, x_2)) = x_1
7.43/2.78	   POL(U22(x_1, x_2)) = x_1
7.43/2.78	   POL(U23(x_1)) = x_1
7.43/2.78	   POL(U31(x_1, x_2)) = x_1
7.43/2.78	   POL(U32(x_1)) = x_1
7.43/2.78	   POL(U41(x_1)) = x_1
7.43/2.78	   POL(U52(x_1, x_2)) = x_1 + x_2
7.43/2.78	   POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.43/2.78	   POL(U62(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.43/2.78	   POL(U63(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.43/2.78	   POL(U64(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.43/2.78	   POL(isNat(x_1)) = 1
7.43/2.78	   POL(isNatKind(x_1)) = 1
7.43/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.43/2.78	   POL(s(x_1)) = 1 + x_1
7.43/2.78	   POL(tt) = 1
7.43/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.43/2.78	
7.43/2.78	   U52(tt, N) -> N
7.43/2.78	
7.43/2.78	
7.43/2.78	
7.43/2.78	
7.43/2.78	----------------------------------------
7.43/2.78	
7.43/2.78	(8)
7.43/2.78	Obligation:
7.43/2.78	Context-sensitive rewrite system:
7.43/2.78	The TRS R consists of the following rules:
7.43/2.78	
7.43/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.43/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.43/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.43/2.78	   U16(tt) -> tt
7.43/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.43/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.43/2.78	   U23(tt) -> tt
7.43/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.43/2.78	   U32(tt) -> tt
7.43/2.78	   U41(tt) -> tt
7.43/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.43/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.43/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.43/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.43/2.78	   isNat(0) -> tt
7.43/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.43/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.43/2.78	   isNatKind(0) -> tt
7.43/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.43/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.43/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.43/2.78	
7.43/2.78	The replacement map contains the following entries:
7.43/2.78	
7.43/2.78	U11: {1}
7.43/2.78	tt: empty set
7.43/2.78	U12: {1}
7.43/2.78	isNatKind: empty set
7.43/2.78	U13: {1}
7.43/2.78	U14: {1}
7.43/2.78	U15: {1}
7.43/2.78	isNat: empty set
7.43/2.78	U16: {1}
7.43/2.78	U21: {1}
7.43/2.78	U22: {1}
7.43/2.78	U23: {1}
7.43/2.78	U31: {1}
7.43/2.78	U32: {1}
7.43/2.78	U41: {1}
7.43/2.78	U61: {1}
7.43/2.78	U62: {1}
7.43/2.78	U63: {1}
7.43/2.78	U64: {1}
7.43/2.78	s: {1}
7.43/2.78	plus: {1, 2}
7.43/2.78	0: empty set
7.43/2.78	
7.43/2.78	----------------------------------------
7.43/2.78	
7.43/2.78	(9) CSRRRRProof (EQUIVALENT)
7.43/2.78	The following CSR is given: Context-sensitive rewrite system:
7.43/2.78	The TRS R consists of the following rules:
7.43/2.78	
7.43/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.43/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.43/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.47/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U61: {1}
7.47/2.78	U62: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 0
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 2*x_1
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = 2*x_1
7.47/2.78	   POL(U15(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U22(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U32(x_1)) = 2*x_1
7.47/2.78	   POL(U41(x_1)) = 2*x_1
7.47/2.78	   POL(U61(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
7.47/2.78	   POL(U62(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
7.47/2.78	   POL(U63(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
7.47/2.78	   POL(U64(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
7.47/2.78	   POL(isNat(x_1)) = 0
7.47/2.78	   POL(isNatKind(x_1)) = 0
7.47/2.78	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
7.47/2.78	   POL(s(x_1)) = 2 + x_1
7.47/2.78	   POL(tt) = 0
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   plus(N, s(M)) -> U61(isNat(M), M, N)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(10)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.47/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U61: {1}
7.47/2.78	U62: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(11) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.47/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U61: {1}
7.47/2.78	U62: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U15(x_1, x_2)) = x_1
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = x_1
7.47/2.78	   POL(U22(x_1, x_2)) = x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U62(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U63(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U64(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(isNat(x_1)) = 1
7.47/2.78	   POL(isNatKind(x_1)) = 1
7.47/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(s(x_1)) = x_1
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U62(tt, M, N) -> U63(isNat(N), M, N)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(12)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U61: {1}
7.47/2.78	U62: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(13) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U61: {1}
7.47/2.78	U62: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U15(x_1, x_2)) = x_1
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = x_1
7.47/2.78	   POL(U22(x_1, x_2)) = x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(U61(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U62(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U63(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U64(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(isNat(x_1)) = 0
7.47/2.78	   POL(isNatKind(x_1)) = 0
7.47/2.78	   POL(plus(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(s(x_1)) = x_1
7.47/2.78	   POL(tt) = 0
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U61(tt, M, N) -> U62(isNatKind(M), M, N)
7.47/2.78	   U64(tt, M, N) -> s(plus(N, M))
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(14)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(15) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	U63: {1}
7.47/2.78	U64: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 0
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 2*x_1
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = 2*x_1
7.47/2.78	   POL(U15(x_1, x_2)) = x_1
7.47/2.78	   POL(U16(x_1)) = 2*x_1
7.47/2.78	   POL(U21(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U22(x_1, x_2)) = x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = 2*x_1
7.47/2.78	   POL(U63(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U64(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(isNat(x_1)) = 0
7.47/2.78	   POL(isNatKind(x_1)) = 0
7.47/2.78	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
7.47/2.78	   POL(s(x_1)) = 2*x_1
7.47/2.78	   POL(tt) = 0
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U63(tt, M, N) -> U64(isNatKind(N), M, N)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(16)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(17) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(0) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U15(x_1, x_2)) = x_1 + 2*x_2
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = 2*x_1 + 2*x_2
7.47/2.78	   POL(U22(x_1, x_2)) = x_1 + 2*x_2
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = 2*x_1
7.47/2.78	   POL(U32(x_1)) = 2*x_1
7.47/2.78	   POL(U41(x_1)) = 2*x_1
7.47/2.78	   POL(isNat(x_1)) = 2*x_1
7.47/2.78	   POL(isNatKind(x_1)) = 0
7.47/2.78	   POL(plus(x_1, x_2)) = 2*x_1 + x_2
7.47/2.78	   POL(s(x_1)) = x_1
7.47/2.78	   POL(tt) = 0
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   isNat(0) -> tt
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(18)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(19) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U15(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(U22(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNat(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 0
7.47/2.78	   POL(plus(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(s(x_1)) = 1 + x_1
7.47/2.78	   POL(tt) = 0
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U21(tt, V1) -> U22(isNatKind(V1), V1)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(20)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(21) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U22: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U15(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U22(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNat(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 0
7.47/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(s(x_1)) = x_1
7.47/2.78	   POL(tt) = 0
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U22(tt, V1) -> U23(isNat(V1))
7.47/2.78	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(22)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(23) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U21: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U15(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U21(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNat(x_1)) = 1 + x_1
7.47/2.78	   POL(isNatKind(x_1)) = 1
7.47/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(s(x_1)) = 1 + x_1
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U15(tt, V2) -> U16(isNat(V2))
7.47/2.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(24)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(25) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U14: {1}
7.47/2.78	U15: {1}
7.47/2.78	isNat: empty set
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
7.47/2.78	   POL(U15(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNat(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 1
7.47/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(s(x_1)) = 1 + x_1
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
7.47/2.78	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(26)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(27) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U13: {1}
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	s: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 0
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_3
7.47/2.78	   POL(U13(x_1, x_2, x_3)) = x_1
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(U32(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 1 + x_1
7.47/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(s(x_1)) = 1 + x_1
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
7.47/2.78	   U31(tt, V2) -> U32(isNatKind(V2))
7.47/2.78	   isNatKind(s(V1)) -> U41(isNatKind(V1))
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(28)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(29) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U31: {1}
7.47/2.78	U32: {1}
7.47/2.78	U41: {1}
7.47/2.78	plus: {1, 2}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 0
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_3
7.47/2.78	   POL(U16(x_1)) = x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U31(x_1, x_2)) = x_1 + x_2
7.47/2.78	   POL(U32(x_1)) = 1 + x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 1 + x_1
7.47/2.78	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U32(tt) -> tt
7.47/2.78	   isNatKind(plus(V1, V2)) -> U31(isNatKind(V1), V2)
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(30)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U41: {1}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(31) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U16(tt) -> tt
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U16: {1}
7.47/2.78	U23: {1}
7.47/2.78	U41: {1}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U16(x_1)) = 1 + x_1
7.47/2.78	   POL(U23(x_1)) = x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 1
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U16(tt) -> tt
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(32)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U23: {1}
7.47/2.78	U41: {1}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(33) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U23(tt) -> tt
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U23: {1}
7.47/2.78	U41: {1}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 1
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
7.47/2.78	   POL(U23(x_1)) = 1 + x_1
7.47/2.78	   POL(U41(x_1)) = x_1
7.47/2.78	   POL(isNatKind(x_1)) = 1
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U23(tt) -> tt
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(34)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U41: {1}
7.47/2.78	0: empty set
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(35) CSRRRRProof (EQUIVALENT)
7.47/2.78	The following CSR is given: Context-sensitive rewrite system:
7.47/2.78	The TRS R consists of the following rules:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	The replacement map contains the following entries:
7.47/2.78	
7.47/2.78	U11: {1}
7.47/2.78	tt: empty set
7.47/2.78	U12: {1}
7.47/2.78	isNatKind: empty set
7.47/2.78	U41: {1}
7.47/2.78	0: empty set
7.47/2.78	Used ordering:
7.47/2.78	Polynomial interpretation [POLO]:
7.47/2.78	
7.47/2.78	   POL(0) = 2
7.47/2.78	   POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U12(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
7.47/2.78	   POL(U41(x_1)) = 2*x_1
7.47/2.78	   POL(isNatKind(x_1)) = 2
7.47/2.78	   POL(tt) = 1
7.47/2.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.47/2.78	
7.47/2.78	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
7.47/2.78	   U41(tt) -> tt
7.47/2.78	   isNatKind(0) -> tt
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(36)
7.47/2.78	Obligation:
7.47/2.78	Context-sensitive rewrite system:
7.47/2.78	R is empty.
7.47/2.78	
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(37) RisEmptyProof (EQUIVALENT)
7.47/2.78	The CSR R is empty. Hence, termination is trivially proven.
7.47/2.78	----------------------------------------
7.47/2.78	
7.47/2.78	(38)
7.47/2.78	YES
7.56/2.89	EOF