Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSToCSRProof (⇔)
↳2 CSR
↳3 CSDependencyPairsProof (⇔)
↳4 QCSDP
↳5 QCSDependencyGraphProof (⇔)
↳6 AND
  ↳7 QCSDP
    ↳8 QCSDPSubtermProof (⇔)
    ↳9 QCSDP
    ↳10 QCSDependencyGraphProof (⇔)
    ↳11 TRUE
  ↳12 QCSDP
    ↳13 QCSUsableRulesProof (⇔)
    ↳14 QCSDP
    ↳15 QCSDPMuMonotonicPoloProof (⇔)
    ↳16 QCSDP
    ↳17 QCSDependencyGraphProof (⇔)
    ↳18 TRUE
  ↳19 QCSDP
    ↳20 QCSDPSubtermProof (⇔)
    ↳21 QCSDP
    ↳22 QCSDependencyGraphProof (⇔)
    ↳23 TRUE
  ↳24 QCSDP
    ↳25 QCSDPSubtermProof (⇔)
    ↳26 QCSDP
    ↳27 QCSDependencyGraphProof (⇔)
    ↳28 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U13(X1, X2, X3)) → U13(active(X1), X2, X3)
active(U14(X1, X2, X3)) → U14(active(X1), X2, X3)
active(U15(X1, X2)) → U15(active(X1), X2)
active(U16(X)) → U16(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2, X3)) → U32(active(X1), X2, X3)
active(U33(X1, X2, X3)) → U33(active(X1), X2, X3)
active(U34(X1, X2, X3)) → U34(active(X1), X2, X3)
active(U35(X1, X2)) → U35(active(X1), X2)
active(U36(X)) → U36(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X)) → U51(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U13(mark(X1), X2, X3) → mark(U13(X1, X2, X3))
U14(mark(X1), X2, X3) → mark(U14(X1, X2, X3))
U15(mark(X1), X2) → mark(U15(X1, X2))
U16(mark(X)) → mark(U16(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2, X3) → mark(U32(X1, X2, X3))
U33(mark(X1), X2, X3) → mark(U33(X1, X2, X3))
U34(mark(X1), X2, X3) → mark(U34(X1, X2, X3))
U35(mark(X1), X2) → mark(U35(X1, X2))
U36(mark(X)) → mark(U36(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X)) → mark(U51(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
s(mark(X)) → mark(s(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNat(X)) → isNat(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U13(X1, X2, X3)) → U13(proper(X1), proper(X2), proper(X3))
proper(U14(X1, X2, X3)) → U14(proper(X1), proper(X2), proper(X3))
proper(U15(X1, X2)) → U15(proper(X1), proper(X2))
proper(U16(X)) → U16(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2, X3)) → U32(proper(X1), proper(X2), proper(X3))
proper(U33(X1, X2, X3)) → U33(proper(X1), proper(X2), proper(X3))
proper(U34(X1, X2, X3)) → U34(proper(X1), proper(X2), proper(X3))
proper(U35(X1, X2)) → U35(proper(X1), proper(X2))
proper(U36(X)) → U36(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X)) → U51(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(0) → ok(0)
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNat(ok(X)) → ok(isNat(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U13(ok(X1), ok(X2), ok(X3)) → ok(U13(X1, X2, X3))
U14(ok(X1), ok(X2), ok(X3)) → ok(U14(X1, X2, X3))
U15(ok(X1), ok(X2)) → ok(U15(X1, X2))
U16(ok(X)) → ok(U16(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2), ok(X3)) → ok(U32(X1, X2, X3))
U33(ok(X1), ok(X2), ok(X3)) → ok(U33(X1, X2, X3))
U34(ok(X1), ok(X2), ok(X3)) → ok(U34(X1, X2, X3))
U35(ok(X1), ok(X2)) → ok(U35(X1, X2))
U36(ok(X)) → ok(U36(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X)) → ok(U51(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
s(ok(X)) → ok(s(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U13(X1, X2, X3)) → U13(active(X1), X2, X3)
active(U14(X1, X2, X3)) → U14(active(X1), X2, X3)
active(U15(X1, X2)) → U15(active(X1), X2)
active(U16(X)) → U16(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2, X3)) → U32(active(X1), X2, X3)
active(U33(X1, X2, X3)) → U33(active(X1), X2, X3)
active(U34(X1, X2, X3)) → U34(active(X1), X2, X3)
active(U35(X1, X2)) → U35(active(X1), X2)
active(U36(X)) → U36(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X)) → U51(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U13(mark(X1), X2, X3) → mark(U13(X1, X2, X3))
U14(mark(X1), X2, X3) → mark(U14(X1, X2, X3))
U15(mark(X1), X2) → mark(U15(X1, X2))
U16(mark(X)) → mark(U16(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2, X3) → mark(U32(X1, X2, X3))
U33(mark(X1), X2, X3) → mark(U33(X1, X2, X3))
U34(mark(X1), X2, X3) → mark(U34(X1, X2, X3))
U35(mark(X1), X2) → mark(U35(X1, X2))
U36(mark(X)) → mark(U36(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X)) → mark(U51(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
s(mark(X)) → mark(s(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNat(X)) → isNat(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U13(X1, X2, X3)) → U13(proper(X1), proper(X2), proper(X3))
proper(U14(X1, X2, X3)) → U14(proper(X1), proper(X2), proper(X3))
proper(U15(X1, X2)) → U15(proper(X1), proper(X2))
proper(U16(X)) → U16(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2, X3)) → U32(proper(X1), proper(X2), proper(X3))
proper(U33(X1, X2, X3)) → U33(proper(X1), proper(X2), proper(X3))
proper(U34(X1, X2, X3)) → U34(proper(X1), proper(X2), proper(X3))
proper(U35(X1, X2)) → U35(proper(X1), proper(X2))
proper(U36(X)) → U36(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X)) → U51(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(0) → ok(0)
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNat(ok(X)) → ok(isNat(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U13(ok(X1), ok(X2), ok(X3)) → ok(U13(X1, X2, X3))
U14(ok(X1), ok(X2), ok(X3)) → ok(U14(X1, X2, X3))
U15(ok(X1), ok(X2)) → ok(U15(X1, X2))
U16(ok(X)) → ok(U16(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2), ok(X3)) → ok(U32(X1, X2, X3))
U33(ok(X1), ok(X2), ok(X3)) → ok(U33(X1, X2, X3))
U34(ok(X1), ok(X2), ok(X3)) → ok(U34(X1, X2, X3))
U35(ok(X1), ok(X2)) → ok(U35(X1, X2))
U36(ok(X)) → ok(U36(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X)) → ok(U51(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
s(ok(X)) → ok(s(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

U101: {1}
tt: empty set
U102: {1}
isNatKind: empty set
U103: {1}
isNat: empty set
U104: {1}
plus: {1, 2}
x: {1, 2}
U11: {1}
U12: {1}
U13: {1}
U14: {1}
U15: {1}
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U33: {1}
U34: {1}
U35: {1}
U36: {1}
U41: {1}
U42: {1}
U51: {1}
U61: {1}
U62: {1}
U71: {1}
U72: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
s: {1}
U91: {1}
U92: {1}
0: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

The replacement map contains the following entries:

U101: {1}
tt: empty set
U102: {1}
isNatKind: empty set
U103: {1}
isNat: empty set
U104: {1}
plus: {1, 2}
x: {1, 2}
U11: {1}
U12: {1}
U13: {1}
U14: {1}
U15: {1}
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U33: {1}
U34: {1}
U35: {1}
U36: {1}
U41: {1}
U42: {1}
U51: {1}
U61: {1}
U62: {1}
U71: {1}
U72: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
s: {1}
U91: {1}
U92: {1}
0: empty set

(3) CSDependencyPairsProof (EQUIVALENT transformation)

Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.

(4) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, PLUS, X, U16', U23', U36', U42', U62', U92', U51'} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U102', U101', U103', U104', U12', U11', U13', U14', U15', U22', U21', U32', U31', U33', U34', U35', U41', U61', U72', U71', U82', U81', U83', U84', U91'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNATKIND, ISNAT, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DPo are:

U101'(tt, M, N) → U102'(isNatKind(M), M, N)
U101'(tt, M, N) → ISNATKIND(M)
U102'(tt, M, N) → U103'(isNat(N), M, N)
U102'(tt, M, N) → ISNAT(N)
U103'(tt, M, N) → U104'(isNatKind(N), M, N)
U103'(tt, M, N) → ISNATKIND(N)
U104'(tt, M, N) → PLUS(x(N, M), N)
U104'(tt, M, N) → X(N, M)
U11'(tt, V1, V2) → U12'(isNatKind(V1), V1, V2)
U11'(tt, V1, V2) → ISNATKIND(V1)
U12'(tt, V1, V2) → U13'(isNatKind(V2), V1, V2)
U12'(tt, V1, V2) → ISNATKIND(V2)
U13'(tt, V1, V2) → U14'(isNatKind(V2), V1, V2)
U13'(tt, V1, V2) → ISNATKIND(V2)
U14'(tt, V1, V2) → U15'(isNat(V1), V2)
U14'(tt, V1, V2) → ISNAT(V1)
U15'(tt, V2) → U16'(isNat(V2))
U15'(tt, V2) → ISNAT(V2)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U21'(tt, V1) → ISNATKIND(V1)
U22'(tt, V1) → U23'(isNat(V1))
U22'(tt, V1) → ISNAT(V1)
U31'(tt, V1, V2) → U32'(isNatKind(V1), V1, V2)
U31'(tt, V1, V2) → ISNATKIND(V1)
U32'(tt, V1, V2) → U33'(isNatKind(V2), V1, V2)
U32'(tt, V1, V2) → ISNATKIND(V2)
U33'(tt, V1, V2) → U34'(isNatKind(V2), V1, V2)
U33'(tt, V1, V2) → ISNATKIND(V2)
U34'(tt, V1, V2) → U35'(isNat(V1), V2)
U34'(tt, V1, V2) → ISNAT(V1)
U35'(tt, V2) → U36'(isNat(V2))
U35'(tt, V2) → ISNAT(V2)
U41'(tt, V2) → U42'(isNatKind(V2))
U41'(tt, V2) → ISNATKIND(V2)
U61'(tt, V2) → U62'(isNatKind(V2))
U61'(tt, V2) → ISNATKIND(V2)
U71'(tt, N) → U72'(isNatKind(N), N)
U71'(tt, N) → ISNATKIND(N)
U81'(tt, M, N) → U82'(isNatKind(M), M, N)
U81'(tt, M, N) → ISNATKIND(M)
U82'(tt, M, N) → U83'(isNat(N), M, N)
U82'(tt, M, N) → ISNAT(N)
U83'(tt, M, N) → U84'(isNatKind(N), M, N)
U83'(tt, M, N) → ISNATKIND(N)
U84'(tt, M, N) → PLUS(N, M)
U91'(tt, N) → U92'(isNatKind(N))
U91'(tt, N) → ISNATKIND(N)
ISNAT(plus(V1, V2)) → U11'(isNatKind(V1), V1, V2)
ISNAT(plus(V1, V2)) → ISNATKIND(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNAT(x(V1, V2)) → U31'(isNatKind(V1), V1, V2)
ISNAT(x(V1, V2)) → ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) → U41'(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → U51'(isNatKind(V1))
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → U61'(isNatKind(V1), V2)
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
PLUS(N, 0) → U71'(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U81'(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U91'(isNat(N), N)
X(N, 0) → ISNAT(N)
X(N, s(M)) → U101'(isNat(M), M, N)
X(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DPc:

U104'(tt, M, N) → N
U104'(tt, M, N) → M
U72'(tt, N) → N
U84'(tt, M, N) → N
U84'(tt, M, N) → M


The hidden terms of R are:
none

Every hiding context is built from:none

Hence, the new unhiding pairs DPu are :

U104'(tt, M, N) → U(N)
U104'(tt, M, N) → U(M)
U72'(tt, N) → U(N)
U84'(tt, M, N) → U(N)
U84'(tt, M, N) → U(M)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(5) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 38 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U41', U61'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

U41'(tt, V2) → ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) → U41'(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → U61'(isNatKind(V1), V2)
U61'(tt, V2) → ISNATKIND(V2)
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(8) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(plus(V1, V2)) → U41'(isNatKind(V1), V2)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → U61'(isNatKind(V1), V2)
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
The remaining pairs can at least be oriented weakly.

U41'(tt, V2) → ISNATKIND(V2)
U61'(tt, V2) → ISNATKIND(V2)
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  x1
U41'(x1, x2)  =  x2
U61'(x1, x2)  =  x2

Subterm Order

(9) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U41', U61'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

U41'(tt, V2) → ISNATKIND(V2)
U61'(tt, V2) → ISNATKIND(V2)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(10) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.

(11) TRUE

(12) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U12', U11', U13', U14', U15', U21', U22', U31', U32', U33', U34', U35'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U11'(tt, V1, V2) → U12'(isNatKind(V1), V1, V2)
U12'(tt, V1, V2) → U13'(isNatKind(V2), V1, V2)
U13'(tt, V1, V2) → U14'(isNatKind(V2), V1, V2)
U14'(tt, V1, V2) → U15'(isNat(V1), V2)
U15'(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11'(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U22'(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(isNatKind(V1), V1, V2)
U31'(tt, V1, V2) → U32'(isNatKind(V1), V1, V2)
U32'(tt, V1, V2) → U33'(isNatKind(V2), V1, V2)
U33'(tt, V1, V2) → U34'(isNatKind(V2), V1, V2)
U34'(tt, V1, V2) → U35'(isNat(V1), V2)
U35'(tt, V2) → ISNAT(V2)
U34'(tt, V1, V2) → ISNAT(V1)
U14'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(13) QCSUsableRulesProof (EQUIVALENT transformation)

The following rules are not useable [DA_EMMES] and can be deleted:

U101(tt, x0, x1) → U102(isNatKind(x0), x0, x1)
U102(tt, x0, x1) → U103(isNat(x1), x0, x1)
U103(tt, x0, x1) → U104(isNatKind(x1), x0, x1)
U104(tt, x0, x1) → plus(x(x1, x0), x1)
U71(tt, x0) → U72(isNatKind(x0), x0)
U72(tt, x0) → x0
U81(tt, x0, x1) → U82(isNatKind(x0), x0, x1)
U82(tt, x0, x1) → U83(isNat(x1), x0, x1)
U83(tt, x0, x1) → U84(isNatKind(x1), x0, x1)
U84(tt, x0, x1) → s(plus(x1, x0))
U91(tt, x0) → U92(isNatKind(x0))
U92(tt) → 0
plus(x0, 0) → U71(isNat(x0), x0)
plus(x0, s(x1)) → U81(isNat(x1), x1, x0)
x(x0, 0) → U91(isNat(x0), x0)
x(x0, s(x1)) → U101(isNat(x1), x1, x0)

(14) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U51, x, U62, U42, U23, U36, U16} are replacing on all positions.
For all symbols f in {U41, U61, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U12', U11', U13', U14', U15', U21', U22', U31', U32', U33', U34', U35'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U11'(tt, V1, V2) → U12'(isNatKind(V1), V1, V2)
U12'(tt, V1, V2) → U13'(isNatKind(V2), V1, V2)
U13'(tt, V1, V2) → U14'(isNatKind(V2), V1, V2)
U14'(tt, V1, V2) → U15'(isNat(V1), V2)
U15'(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11'(isNatKind(V1), V1, V2)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U22'(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(isNatKind(V1), V1, V2)
U31'(tt, V1, V2) → U32'(isNatKind(V1), V1, V2)
U32'(tt, V1, V2) → U33'(isNatKind(V2), V1, V2)
U33'(tt, V1, V2) → U34'(isNatKind(V2), V1, V2)
U34'(tt, V1, V2) → U35'(isNat(V1), V2)
U35'(tt, V2) → ISNAT(V2)
U34'(tt, V1, V2) → ISNAT(V1)
U14'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U51(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

(15) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

By using the following µ-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
Strictly oriented dependency pairs:

ISNAT(plus(V1, V2)) → U11'(isNatKind(V1), V1, V2)
U22'(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(isNatKind(V1), V1, V2)
U31'(tt, V1, V2) → U32'(isNatKind(V1), V1, V2)


Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 1   
POL(ISNAT(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U11'(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(U12(x1, x2, x3)) = x1   
POL(U12'(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(U13(x1, x2, x3)) = 2·x1   
POL(U13'(x1, x2, x3)) = x1 + x2 + x3   
POL(U14(x1, x2, x3)) = x1   
POL(U14'(x1, x2, x3)) = x1 + x2 + x3   
POL(U15(x1, x2)) = x1   
POL(U15'(x1, x2)) = 2·x1 + x2   
POL(U16(x1)) = 2·x1   
POL(U21(x1, x2)) = 2·x1   
POL(U21'(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U22(x1, x2)) = 2·x1   
POL(U22'(x1, x2)) = 1 + 2·x1 + x2   
POL(U23(x1)) = 2·x1   
POL(U31(x1, x2, x3)) = x1   
POL(U31'(x1, x2, x3)) = 1 + 2·x1 + x2 + 2·x3   
POL(U32(x1, x2, x3)) = x1   
POL(U32'(x1, x2, x3)) = 2·x1 + x2 + 2·x3   
POL(U33(x1, x2, x3)) = x1   
POL(U33'(x1, x2, x3)) = x1 + x2 + 2·x3   
POL(U34(x1, x2, x3)) = x1   
POL(U34'(x1, x2, x3)) = x1 + x2 + 2·x3   
POL(U35(x1, x2)) = 2·x1   
POL(U35'(x1, x2)) = 2·x1 + 2·x2   
POL(U36(x1)) = 2·x1   
POL(U41(x1, x2)) = x1   
POL(U42(x1)) = 2·x1   
POL(U51(x1)) = x1   
POL(U61(x1, x2)) = 2·x1   
POL(U62(x1)) = 2·x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(plus(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(s(x1)) = 1 + 2·x1   
POL(tt) = 0   
POL(x(x1, x2)) = 2 + 2·x1 + 2·x2   

(16) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, s, U51, x, U62, U42, U23, U36, U16} are replacing on all positions.
For all symbols f in {U41, U61, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U12', U11', U13', U14', U15', U21', U22', U33', U32', U34', U35'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U11'(tt, V1, V2) → U12'(isNatKind(V1), V1, V2)
U12'(tt, V1, V2) → U13'(isNatKind(V2), V1, V2)
U13'(tt, V1, V2) → U14'(isNatKind(V2), V1, V2)
U14'(tt, V1, V2) → U15'(isNat(V1), V2)
U15'(tt, V2) → ISNAT(V2)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U32'(tt, V1, V2) → U33'(isNatKind(V2), V1, V2)
U33'(tt, V1, V2) → U34'(isNatKind(V2), V1, V2)
U34'(tt, V1, V2) → U35'(isNat(V1), V2)
U35'(tt, V2) → ISNAT(V2)
U34'(tt, V1, V2) → ISNAT(V1)
U14'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U51(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U23(tt) → tt
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt

Q is empty.

(17) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 13 less nodes.

(18) TRUE

(19) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, PLUS} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U83', U82', U84', U81'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U82'(tt, M, N) → U83'(isNat(N), M, N)
U83'(tt, M, N) → U84'(isNatKind(N), M, N)
U84'(tt, M, N) → PLUS(N, M)
PLUS(N, s(M)) → U81'(isNat(M), M, N)
U81'(tt, M, N) → U82'(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(20) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.


PLUS(N, s(M)) → U81'(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

U82'(tt, M, N) → U83'(isNat(N), M, N)
U83'(tt, M, N) → U84'(isNatKind(N), M, N)
U84'(tt, M, N) → PLUS(N, M)
U81'(tt, M, N) → U82'(isNatKind(M), M, N)
Used ordering: Combined order from the following AFS and order.
U83'(x1, x2, x3)  =  x2
U82'(x1, x2, x3)  =  x2
U84'(x1, x2, x3)  =  x2
PLUS(x1, x2)  =  x2
U81'(x1, x2, x3)  =  x2

Subterm Order

(21) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, PLUS} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U83', U82', U84', U81'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U82'(tt, M, N) → U83'(isNat(N), M, N)
U83'(tt, M, N) → U84'(isNatKind(N), M, N)
U84'(tt, M, N) → PLUS(N, M)
U81'(tt, M, N) → U82'(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(22) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 4 less nodes.

(23) TRUE

(24) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, X} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U103', U102', U104', U101'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U102'(tt, M, N) → U103'(isNat(N), M, N)
U103'(tt, M, N) → U104'(isNatKind(N), M, N)
U104'(tt, M, N) → X(N, M)
X(N, s(M)) → U101'(isNat(M), M, N)
U101'(tt, M, N) → U102'(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(25) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.


X(N, s(M)) → U101'(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

U102'(tt, M, N) → U103'(isNat(N), M, N)
U103'(tt, M, N) → U104'(isNatKind(N), M, N)
U104'(tt, M, N) → X(N, M)
U101'(tt, M, N) → U102'(isNatKind(M), M, N)
Used ordering: Combined order from the following AFS and order.
U103'(x1, x2, x3)  =  x2
U102'(x1, x2, x3)  =  x2
U104'(x1, x2, x3)  =  x2
X(x1, x2)  =  x2
U101'(x1, x2, x3)  =  x2

Subterm Order

(26) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {plus, x, U16, U23, U36, U42, U51, U62, s, U92, X} are replacing on all positions.
For all symbols f in {U101, U102, U103, U104, U11, U12, U13, U14, U15, U21, U22, U31, U32, U33, U34, U35, U41, U61, U71, U72, U81, U82, U83, U84, U91, U103', U102', U104', U101'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat} are not replacing on any position.

The TRS P consists of the following rules:

U102'(tt, M, N) → U103'(isNat(N), M, N)
U103'(tt, M, N) → U104'(isNatKind(N), M, N)
U104'(tt, M, N) → X(N, M)
U101'(tt, M, N) → U102'(isNatKind(M), M, N)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(M), M, N)
U102(tt, M, N) → U103(isNat(N), M, N)
U103(tt, M, N) → U104(isNatKind(N), M, N)
U104(tt, M, N) → plus(x(N, M), N)
U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(V1), V1, V2)
U32(tt, V1, V2) → U33(isNatKind(V2), V1, V2)
U33(tt, V1, V2) → U34(isNatKind(V2), V1, V2)
U34(tt, V1, V2) → U35(isNat(V1), V2)
U35(tt, V2) → U36(isNat(V2))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(V2))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(V2))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(N), N)
U72(tt, N) → N
U81(tt, M, N) → U82(isNatKind(M), M, N)
U82(tt, M, N) → U83(isNat(N), M, N)
U83(tt, M, N) → U84(isNatKind(N), M, N)
U84(tt, M, N) → s(plus(N, M))
U91(tt, N) → U92(isNatKind(N))
U92(tt) → 0
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U41(isNatKind(V1), V2)
isNatKind(s(V1)) → U51(isNatKind(V1))
isNatKind(x(V1, V2)) → U61(isNatKind(V1), V2)
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)

Q is empty.

(27) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 4 less nodes.

(28) TRUE