Termination w.r.t. Q proof of /tmp/tmpl7hLqX/PEANO_complete_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 DependencyGraphProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 DependencyGraphProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 TRUE

  ↳19 QDP

(0) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__ISNATKIND(V1)
A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U12(tt, V1, V2) → A__ISNATKIND(V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__ISNATKIND(V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U14(tt, V1, V2) → A__ISNAT(V1)
A__U15(tt, V2) → A__U16(a__isNat(V2))
A__U15(tt, V2) → A__ISNAT(V2)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNATKIND(V1)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U22(tt, V1) → A__ISNAT(V1)
A__U31(tt, V2) → A__U32(a__isNatKind(V2))
A__U31(tt, V2) → A__ISNATKIND(V2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U51(tt, N) → A__ISNATKIND(N)
A__U52(tt, N) → MARK(N)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U61(tt, M, N) → A__ISNATKIND(M)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U62(tt, M, N) → A__ISNAT(N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U63(tt, M, N) → A__ISNATKIND(N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U64(tt, M, N) → MARK(N)
A__U64(tt, M, N) → MARK(M)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__U41(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U13(X1, X2, X3)) → A__U13(mark(X1), X2, X3)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → A__U14(mark(X1), X2, X3)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → A__U15(mark(X1), X2)
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U16(X)) → A__U16(mark(X))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → A__U23(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → A__U41(mark(X))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 30 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U31(x1, x2) = x2
tt = tt
A__ISNATKIND(x1) = x1
plus(x1, x2) = plus(x1, x2)
a__isNatKind(x1) = a__isNatKind
s(x1) = s(x1)
0 = 0
a__U31(x1, x2) = a__U31
a__U41(x1) = x1
isNatKind(x1) = isNatKind(x1)
U31(x1, x2) = U31
U41(x1) = x1
a__U32(x1) = a__U32(x1)
U32(x1) = x1

Recursive path order with status [RPO].
Precedence:

plus₂ > a_{_U31} > a_{_isNatKind}
s₁ > a_{_isNatKind}
0 > a_{_isNatKind}
isNatKind₁ > a_{_isNatKind}
U31 > a_{_isNatKind}
a_{_U32₁} > tt > a_{_isNatKind}

Status:

tt: multiset
plus₂: multiset
a_{_isNatKind}: multiset
s₁: multiset
0: multiset
a_{_U31}: multiset
isNatKind₁: multiset
U31: multiset
a_{_U32₁}: multiset

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U31(tt, V2) → A__ISNATKIND(V2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
A__U14(tt, V1, V2) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U12(x1, x2, x3) = A__U12(x2, x3)
tt = tt
A__U13(x1, x2, x3) = A__U13(x1, x2, x3)
a__isNatKind(x1) = a__isNatKind(x1)
A__U14(x1, x2, x3) = A__U14(x2, x3)
A__U15(x1, x2) = x2
a__isNat(x1) = a__isNat(x1)
A__ISNAT(x1) = x1
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x2, x3)
s(x1) = x1
A__U21(x1, x2) = x2
A__U22(x1, x2) = x2
0 = 0
a__U31(x1, x2) = a__U31(x1, x2)
a__U41(x1) = x1
isNatKind(x1) = x1
a__U11(x1, x2, x3) = a__U11
a__U21(x1, x2) = a__U21(x1, x2)
isNat(x1) = x1
a__U12(x1, x2, x3) = a__U12
a__U13(x1, x2, x3) = x3
U13(x1, x2, x3) = U13
a__U14(x1, x2, x3) = x3
U14(x1, x2, x3) = x3
a__U15(x1, x2) = x2
U15(x1, x2) = x2
U11(x1, x2, x3) = U11
U12(x1, x2, x3) = U12
U21(x1, x2) = x2
a__U22(x1, x2) = a__U22
U22(x1, x2) = x2
a__U23(x1) = a__U23(x1)
U23(x1) = U23(x1)
a__U16(x1) = a__U16(x1)
U16(x1) = U16(x1)
U31(x1, x2) = x2
U41(x1) = x1
a__U32(x1) = x1
U32(x1) = x1

Recursive path order with status [RPO].
Precedence:

a_{_isNat₁} > tt > a_{_isNatKind₁}
a_{_isNat₁} > a_{_U11} > a_{_isNatKind₁}
a_{_isNat₁} > a_{_U21₂} > a_{_isNatKind₁}
plus₂ > A_{_U11₂} > A_{_U12₂} > A_{_U13₃} > A_{_U14₂} > a_{_isNatKind₁}
plus₂ > a_{_U31₂} > a_{_isNatKind₁}
plus₂ > a_{_U11} > a_{_isNatKind₁}
0 > tt > a_{_isNatKind₁}
a_{_U12} > a_{_isNatKind₁}
U13 > a_{_isNatKind₁}
U11 > a_{_isNatKind₁}
U12 > a_{_isNatKind₁}
a_{_U22} > a_{_isNatKind₁}
a_{_U23₁} > tt > a_{_isNatKind₁}
U23₁ > a_{_isNatKind₁}
a_{_U16₁} > tt > a_{_isNatKind₁}
U16₁ > a_{_isNatKind₁}

Status:

A_{_U12₂}: multiset
tt: multiset
A_{_U13₃}: multiset
a_{_isNatKind₁}: multiset
A_{_U14₂}: multiset
a_{_isNat₁}: multiset
plus₂: multiset
A_{_U11₂}: multiset
0: multiset
a_{_U31₂}: multiset
a_{_U11}: multiset
a_{_U21₂}: multiset
a_{_U12}: multiset
U13: multiset
U11: multiset
U12: multiset
a_{_U22}: multiset
a_{_U23₁}: multiset
U23₁: multiset
a_{_U16₁}: multiset
U16₁: multiset

The following usable rules [FROCOS05] were oriented:

a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__isNatKind(X) → isNatKind(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(tt) → tt
a__U41(X) → U41(X)
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U32(X) → U32(X)

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U15(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U21(x1, x2) = x2
tt = tt
A__U22(x1, x2) = x2
a__isNatKind(x1) = a__isNatKind
A__ISNAT(x1) = x1
s(x1) = s(x1)
0 = 0
plus(x1, x2) = plus(x1, x2)
a__U31(x1, x2) = a__U31(x1)
a__U41(x1) = x1
isNatKind(x1) = isNatKind(x1)
U31(x1, x2) = U31
U41(x1) = U41(x1)
a__U32(x1) = a__U32(x1)
U32(x1) = U32

Recursive path order with status [RPO].
Precedence:

s₁ > a_{_isNatKind}
0 > tt > a_{_isNatKind}
plus₂ > a_{_U31₁} > a_{_isNatKind}
isNatKind₁ > a_{_isNatKind}
U31 > a_{_isNatKind}
U41₁ > a_{_isNatKind}
a_{_U32₁} > tt > a_{_isNatKind}
U32 > a_{_isNatKind}

Status:

tt: multiset
a_{_isNatKind}: multiset
s₁: multiset
0: multiset
plus₂: multiset
a_{_U31₁}: multiset
isNatKind₁: multiset
U31: multiset
U41₁: multiset
a_{_U32₁}: multiset
U32: multiset

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(18) TRUE

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)
A__U52(tt, N) → MARK(N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
A__U64(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U51(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U64(tt, M, N) → MARK(N)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
A__U64(tt, M, N) → MARK(M)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__U13(tt, V1, V2) → a__U14(a__isNatKind(V2), V1, V2)
a__U14(tt, V1, V2) → a__U15(a__isNat(V1), V2)
a__U15(tt, V2) → a__U16(a__isNat(V2))
a__U16(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U32(tt) → tt
a__U41(tt) → tt
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__U52(tt, N) → mark(N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))
a__plus(N, 0) → a__U51(a__isNat(N), N)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U15(X1, X2) → U15(X1, X2)
a__isNat(X) → isNat(X)
a__U16(X) → U16(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X) → U41(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.