Termination w.r.t. Q proof of /tmp/tmpAsCOx5/PEANO_complete_noand

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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.

↳ QTRS

  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 30 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V2) → A__ISNATKIND(V2)
A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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__U41(tt) → tt
a__U41(X) → U41(X)
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

A__ISNATKIND(plus(V1, V2)) → A__U31(a__isNatKind(V1), V2)
The graph contains the following edges 1 > 2
A__U31(tt, V2) → A__ISNATKIND(V2)
The graph contains the following edges 2 >= 1
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPSizeChangeProof

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

A__U13(tt, V1, V2) → A__U14(a__isNatKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U11(tt, V1, V2) → A__U12(a__isNatKind(V1), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U12(tt, V1, V2) → A__U13(a__isNatKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U15(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
A__U14(tt, V1, V2) → A__U15(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2
A__U14(tt, V1, V2) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U22(tt, V1) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
The graph contains the following edges 2 >= 2
A__ISNAT(plus(V1, V2)) → A__U11(a__isNatKind(V1), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
The graph contains the following edges 1 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → MARK(X)
A__PLUS(N, s(M)) → A__U61(a__isNat(M), M, N)
A__U51(tt, N) → A__U52(a__isNatKind(N), N)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__PLUS(x₁, x₂)) = x₁ + x₂
POL(A__U51(x₁, x₂)) = x₁ + x₂
POL(A__U52(x₁, x₂)) = x₂
POL(A__U61(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U62(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U63(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U64(x₁, x₂, x₃)) = x₂ + x₃
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = x₁
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂, x₃)) = x₁
POL(a__U13(x₁, x₂, x₃)) = x₁
POL(a__U14(x₁, x₂, x₃)) = x₁
POL(a__U15(x₁, x₂)) = x₁
POL(a__U16(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁, x₂)) = x₁ + x₂
POL(a__U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 1

The following usable rules [17] were oriented:

a__U16(X) → U16(X)
a__isNat(X) → isNat(X)
a__U15(X1, X2) → U15(X1, X2)
a__U14(X1, X2, X3) → U14(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U23(X) → U23(X)
a__U22(X1, X2) → U22(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2) → U51(X1, X2)
a__U41(X) → U41(X)
a__U32(X) → U32(X)
a__U64(X1, X2, X3) → U64(X1, X2, X3)
a__U63(X1, X2, X3) → U63(X1, X2, X3)
a__U62(X1, X2, X3) → U62(X1, X2, X3)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
a__U52(tt, N) → mark(N)
a__U51(tt, N) → a__U52(a__isNatKind(N), N)
a__plus(N, 0) → a__U51(a__isNat(N), N)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X)) → a__U41(mark(X))
mark(U63(X1, X2, X3)) → a__U63(mark(X1), X2, X3)
mark(U64(X1, X2, X3)) → a__U64(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2, X3)) → a__U62(mark(X1), X2, X3)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__U13(X1, X2, X3) → U13(X1, X2, X3)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
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__U11(tt, V1, V2) → a__U12(a__isNatKind(V1), V1, V2)
a__U12(tt, V1, V2) → a__U13(a__isNatKind(V2), V1, V2)
a__isNatKind(0) → tt
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(plus(V1, V2)) → a__U11(a__isNatKind(V1), V1, V2)
a__isNat(0) → tt
a__U64(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U63(tt, M, N) → a__U64(a__isNatKind(N), M, N)
a__U62(tt, M, N) → a__U63(a__isNat(N), M, N)
a__U61(tt, M, N) → a__U62(a__isNatKind(M), M, N)
a__U41(tt) → tt
a__U32(tt) → tt
a__U31(tt, V2) → a__U32(a__isNatKind(V2))
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U16(X)) → a__U16(mark(X))
mark(U14(X1, X2, X3)) → a__U14(mark(X1), X2, X3)
mark(U15(X1, X2)) → a__U15(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U13(X1, X2, X3)) → a__U13(mark(X1), X2, X3)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U61(a__isNat(M), M, N)
a__isNatKind(plus(V1, V2)) → a__U31(a__isNatKind(V1), V2)
a__isNatKind(s(V1)) → a__U41(a__isNatKind(V1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPSizeChangeProof

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

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

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)

From the DPs we obtained the following set of size-change graphs:

A__U52(tt, N) → MARK(N)
The graph contains the following edges 2 >= 1
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
The graph contains the following edges 1 > 2
A__U61(tt, M, N) → A__U62(a__isNatKind(M), M, N)
The graph contains the following edges 2 >= 2, 3 >= 3
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
The graph contains the following edges 1 > 2, 1 > 3
A__U62(tt, M, N) → A__U63(a__isNat(N), M, N)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U63(tt, M, N) → A__U64(a__isNatKind(N), M, N)
The graph contains the following edges 2 >= 2, 3 >= 3
MARK(U62(X1, X2, X3)) → A__U62(mark(X1), X2, X3)
The graph contains the following edges 1 > 2, 1 > 3
MARK(U63(X1, X2, X3)) → A__U63(mark(X1), X2, X3)
The graph contains the following edges 1 > 2, 1 > 3
MARK(U64(X1, X2, X3)) → A__U64(mark(X1), X2, X3)
The graph contains the following edges 1 > 2, 1 > 3
MARK(U11(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U12(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U13(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U14(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U15(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U16(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U21(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U22(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U23(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U31(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U32(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U41(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U51(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U52(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U61(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U62(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U63(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U64(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(plus(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(plus(X1, X2)) → MARK(X2)
The graph contains the following edges 1 > 1
A__U64(tt, M, N) → MARK(N)
The graph contains the following edges 3 >= 1
A__U64(tt, M, N) → MARK(M)
The graph contains the following edges 2 >= 1