Termination w.r.t. Q proof of /tmp/tmp4mxWKH/MYNAT_nokinds_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, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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__U72(tt, M, N) → MARK(M)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
MARK(U11(X1, X2)) → MARK(X1)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__ISNAT(N)
A__X(N, 0) → A__U61(a__isNat(N))
A__U52(tt, M, N) → MARK(M)
A__U71(tt, M, N) → A__ISNAT(N)
MARK(x(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U52(tt, M, N) → MARK(N)
A__X(N, 0) → A__ISNAT(N)
MARK(U32(X)) → MARK(X)
A__U41(tt, N) → MARK(N)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, V2) → A__ISNAT(V2)
MARK(U12(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
MARK(s(X)) → MARK(X)
MARK(U12(X)) → A__U12(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X)) → A__U21(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U72(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → MARK(N)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__U11(tt, V2) → A__U12(a__isNat(V2))
A__U31(tt, V2) → A__U32(a__isNat(V2))
MARK(U52(X1, X2, X3)) → MARK(X1)
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
MARK(U21(X)) → MARK(X)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U32(X)) → A__U32(mark(X))
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U61(X)) → A__U61(mark(X))
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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__U72(tt, M, N) → MARK(M)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
MARK(U11(X1, X2)) → MARK(X1)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__ISNAT(N)
A__X(N, 0) → A__U61(a__isNat(N))
A__U52(tt, M, N) → MARK(M)
A__U71(tt, M, N) → A__ISNAT(N)
MARK(x(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U52(tt, M, N) → MARK(N)
A__X(N, 0) → A__ISNAT(N)
MARK(U32(X)) → MARK(X)
A__U41(tt, N) → MARK(N)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, V2) → A__ISNAT(V2)
MARK(U12(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
MARK(s(X)) → MARK(X)
MARK(U12(X)) → A__U12(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X)) → A__U21(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U72(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → MARK(N)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__U11(tt, V2) → A__U12(a__isNat(V2))
A__U31(tt, V2) → A__U32(a__isNat(V2))
MARK(U52(X1, X2, X3)) → MARK(X1)
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
MARK(U21(X)) → MARK(X)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U32(X)) → A__U32(mark(X))
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U61(X)) → A__U61(mark(X))
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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 2 SCCs with 17 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U31(tt, V2) → A__ISNAT(V2)
A__U11(tt, V2) → A__ISNAT(V2)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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

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

A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U11(tt, V2) → A__ISNAT(V2)
A__U31(tt, V2) → A__ISNAT(V2)

The TRS R consists of the following rules:

a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(X) → isNat(X)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt) → tt
a__U12(X) → U12(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__U31(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
A__U11(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__ISNAT(s(V1)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

A__U72(tt, M, N) → MARK(M)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
MARK(U12(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U52(tt, M, N) → MARK(M)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(x(X1, X2)) → MARK(X1)
MARK(U72(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → MARK(N)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U41(tt, N) → MARK(N)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(x(X1, X2)) → MARK(X2)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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.

A__U72(tt, M, N) → MARK(M)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U52(tt, M, N) → MARK(M)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(x(X1, X2)) → MARK(X1)
MARK(U72(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → MARK(N)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → MARK(X1)
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X1)
A__U41(tt, N) → MARK(N)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(x(X1, X2)) → MARK(X2)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

The remaining pairs can at least be oriented weakly.

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)

Used ordering: Combined order from the following AFS and order.
A__U72(x1, x2, x3) = A__U72(x1, x2, x3)
tt = tt
MARK(x1) = MARK(x1)
A__PLUS(x1, x2) = A__PLUS(x1, x2)
s(x1) = s(x1)
A__U51(x1, x2, x3) = A__U51(x1, x2, x3)
a__isNat(x1) = a__isNat
U12(x1) = x1
U71(x1, x2, x3) = U71(x1, x2, x3)
U11(x1, x2) = x1
A__U52(x1, x2, x3) = A__U52(x1, x2, x3)
U51(x1, x2, x3) = U51(x1, x2, x3)
mark(x1) = x1
U31(x1, x2) = x1
plus(x1, x2) = plus(x1, x2)
U41(x1, x2) = U41(x1, x2)
A__U41(x1, x2) = A__U41(x2)
a__x(x1, x2) = a__x(x1, x2)
x(x1, x2) = x(x1, x2)
U72(x1, x2, x3) = U72(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
A__X(x1, x2) = A__X(x1, x2)
0 = 0
U32(x1) = x1
U21(x1) = x1
A__U71(x1, x2, x3) = A__U71(x1, x2, x3)
U61(x1) = U61(x1)
a__U72(x1, x2, x3) = a__U72(x1, x2, x3)
a__plus(x1, x2) = a__plus(x1, x2)
a__U52(x1, x2, x3) = a__U52(x1, x2, x3)
a__U71(x1, x2, x3) = a__U71(x1, x2, x3)
a__U61(x1) = a__U61(x1)
a__U32(x1) = x1
a__U51(x1, x2, x3) = a__U51(x1, x2, x3)
a__U11(x1, x2) = x1
a__U12(x1) = x1
a__U21(x1) = x1
a__U31(x1, x2) = x1
isNat(x1) = isNat
a__U41(x1, x2) = a__U41(x1, x2)

Recursive path order with status [2].
Quasi-Precedence:

[A_{_U72₃}, U71₃, a_{_x₂}, x₂, U72₃, A_{_X₂}, A_{_U71₃}, a_{_U72₃}, a_{_U71₃}] > [U51₃, plus₂, U52₃, a_{_plus₂}, a_{_U52₃}, a_{_U51₃}] > [A_{_PLUS₂}, A_{_U51₃}, A_{_U52₃}] > [tt, s₁, a_{_isNat}, isNat] > 0 > A_{_U41₁} > MARK₁ > [U61₁, a_{_U61₁}]
[A_{_U72₃}, U71₃, a_{_x₂}, x₂, U72₃, A_{_X₂}, A_{_U71₃}, a_{_U72₃}, a_{_U71₃}] > [U51₃, plus₂, U52₃, a_{_plus₂}, a_{_U52₃}, a_{_U51₃}] > [U41₂, a_{_U41₂}] > A_{_U41₁} > MARK₁ > [U61₁, a_{_U61₁}]

Status:

A_{_PLUS₂}: multiset
A_{_U51₃}: multiset
x₂: [1,2]
a_{_plus₂}: multiset
isNat: multiset
a_{_U61₁}: [1]
tt: multiset
A_{_U41₁}: [1]
s₁: multiset
U51₃: multiset
plus₂: multiset
a_{_U41₂}: multiset
U61₁: [1]
a_{_isNat}: multiset
a_{_U52₃}: multiset
0: multiset
a_{_U51₃}: multiset
U52₃: multiset
a_{_x₂}: [1,2]
A_{_U71₃}: [3,2,1]
MARK₁: multiset
A_{_U52₃}: multiset
U41₂: multiset
A_{_X₂}: [1,2]
U72₃: [3,2,1]
A_{_U72₃}: [3,2,1]
a_{_U71₃}: [3,2,1]
a_{_U72₃}: [3,2,1]
U71₃: [3,2,1]

The following usable rules [17] were oriented:

a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__U32(tt) → tt
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__isNat(0) → tt
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U21(X)) → a__U21(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U12(X)) → a__U12(mark(X))
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
mark(U61(X)) → a__U61(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(tt) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
MARK(U31(X1, X2)) → MARK(X1)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
MARK(U21(X)) → MARK(X)
MARK(U32(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ UsableRulesProof

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U21(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U32(X)) → MARK(X)

R is empty.
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:

MARK(U12(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U11(X1, X2)) → MARK(X1)
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(U21(X)) → MARK(X)
The graph contains the following edges 1 > 1