Termination w.r.t. Q proof of /tmp/tmpaf0SY9/MYNAT_complete

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 PisEmptyProof (⇔)

↳14 TRUE

(0) Obligation:

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

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__U12(tt, V2) → A__ISNAT(V2)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U32(tt, V2) → A__U33(a__isNat(V2))
A__U32(tt, V2) → A__ISNAT(V2)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U51(tt, M, N) → MARK(N)
A__U51(tt, M, N) → MARK(M)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U71(tt, M, N) → MARK(N)
A__U71(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(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__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, 0) → A__U61(a__and(a__isNat(N), isNatKind(N)))
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__X(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)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → A__U22(mark(X))
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(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(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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 1 SCC with 8 less nodes.

(4) Obligation:

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

A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
A__U32(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U51(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__X(N, s(M)) → A__ISNAT(M)
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(M)
A__U51(tt, M, N) → MARK(M)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
A__U32(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U51(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__X(N, s(M)) → A__ISNAT(M)
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(M)
A__U51(tt, M, N) → MARK(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U12(x1, x2) = A__U12(x2)
tt = tt
A__ISNAT(x1) = A__ISNAT(x1)
plus(x1, x2) = plus(x1, x2)
A__U11(x1, x2, x3) = A__U11(x2, x3)
a__and(x1, x2) = a__and(x1, x2)
a__isNatKind(x1) = x1
isNatKind(x1) = x1
a__isNat(x1) = x1
A__AND(x1, x2) = A__AND(x2)
MARK(x1) = MARK(x1)
U11(x1, x2, x3) = U11(x1, x2, x3)
mark(x1) = x1
U12(x1, x2) = U12(x1, x2)
isNat(x1) = x1
A__ISNATKIND(x1) = A__ISNATKIND(x1)
s(x1) = s(x1)
x(x1, x2) = x(x1, x2)
A__U21(x1, x2) = A__U21(x2)
A__U31(x1, x2, x3) = A__U31(x1, x2, x3)
A__U32(x1, x2) = A__U32(x2)
U13(x1) = x1
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
U31(x1, x2, x3) = U31(x1, x2, x3)
U32(x1, x2) = U32(x1, x2)
U33(x1) = x1
U41(x1, x2) = U41(x1, x2)
A__U41(x1, x2) = A__U41(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
A__U51(x1, x2, x3) = A__U51(x1, x2, x3)
A__PLUS(x1, x2) = A__PLUS(x1, x2)
0 = 0
and(x1, x2) = and(x1, x2)
U61(x1) = U61(x1)
U71(x1, x2, x3) = U71(x1, x2, x3)
A__U71(x1, x2, x3) = A__U71(x1, x2, x3)
a__x(x1, x2) = a__x(x1, x2)
A__X(x1, x2) = A__X(x1, x2)
a__U41(x1, x2) = a__U41(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
a__U71(x1, x2, x3) = a__U71(x1, x2, x3)
a__U11(x1, x2, x3) = a__U11(x1, x2, x3)
a__U21(x1, x2) = a__U21(x1, x2)
a__U31(x1, x2, x3) = a__U31(x1, x2, x3)
a__U12(x1, x2) = a__U12(x1, x2)
a__U13(x1) = x1
a__U22(x1) = a__U22(x1)
a__U32(x1, x2) = a__U32(x1, x2)
a__U33(x1) = x1
a__U51(x1, x2, x3) = a__U51(x1, x2, x3)
a__U61(x1) = a__U61(x1)

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

[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [plus₂, U51₃, A_{_U51₃}, A_{_PLUS₂}, a_{_plus₂}, a_{_U51₃}] > [A_{_AND₁}, MARK₁, A_{_U41₂}] > A_{_U11₂} > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > [a_{_and₂}, and₂]
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [plus₂, U51₃, A_{_U51₃}, A_{_PLUS₂}, a_{_plus₂}, a_{_U51₃}] > [A_{_AND₁}, MARK₁, A_{_U41₂}] > A_{_U11₂} > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > A_{_ISNATKIND₁}
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [plus₂, U51₃, A_{_U51₃}, A_{_PLUS₂}, a_{_plus₂}, a_{_U51₃}] > [A_{_AND₁}, MARK₁, A_{_U41₂}] > A_{_U11₂} > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > [U21₂, a_{_U21₂}] > [U22₁, a_{_U22₁}] > tt
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [plus₂, U51₃, A_{_U51₃}, A_{_PLUS₂}, a_{_plus₂}, a_{_U51₃}] > [U11₃, a_{_U11₃}] > [U12₂, a_{_U12₂}]
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [plus₂, U51₃, A_{_U51₃}, A_{_PLUS₂}, a_{_plus₂}, a_{_U51₃}] > [U41₂, a_{_U41₂}]
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [U31₃, U32₂, a_{_U31₃}, a_{_U32₂}] > [A_{_AND₁}, MARK₁, A_{_U41₂}] > A_{_U11₂} > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > [a_{_and₂}, and₂]
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [U31₃, U32₂, a_{_U31₃}, a_{_U32₂}] > [A_{_AND₁}, MARK₁, A_{_U41₂}] > A_{_U11₂} > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > A_{_ISNATKIND₁}
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [U31₃, U32₂, a_{_U31₃}, a_{_U32₂}] > [A_{_AND₁}, MARK₁, A_{_U41₂}] > A_{_U11₂} > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > [U21₂, a_{_U21₂}] > [U22₁, a_{_U22₁}] > tt
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [U31₃, U32₂, a_{_U31₃}, a_{_U32₂}] > [A_{_U31₃}, A_{_U32₁}] > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > [a_{_and₂}, and₂]
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [U31₃, U32₂, a_{_U31₃}, a_{_U32₂}] > [A_{_U31₃}, A_{_U32₁}] > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > A_{_ISNATKIND₁}
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [U31₃, U32₂, a_{_U31₃}, a_{_U32₂}] > [A_{_U31₃}, A_{_U32₁}] > [A_{_U12₁}, A_{_ISNAT₁}, s₁, A_{_U21₁}] > [U21₂, a_{_U21₂}] > [U22₁, a_{_U22₁}] > tt
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [0, U61₁, a_{_U61₁}] > tt
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [0, U61₁, a_{_U61₁}] > [a_{_and₂}, and₂]
[x₂, U71₃, A_{_U71₃}, a_{_x₂}, A_{_X₂}, a_{_U71₃}] > [0, U61₁, a_{_U61₁}] > [U41₂, a_{_U41₂}]

Status:

A_{_U12₁}: multiset
tt: multiset
A_{_ISNAT₁}: multiset
plus₂: [2,1]
A_{_U11₂}: multiset
a_{_and₂}: multiset
A_{_AND₁}: multiset
MARK₁: multiset
U11₃: multiset
U12₂: multiset
A_{_ISNATKIND₁}: [1]
s₁: [1]
x₂: [1,2]
A_{_U21₁}: multiset
A_{_U31₃}: multiset
A_{_U32₁}: multiset
U21₂: multiset
U22₁: [1]
U31₃: multiset
U32₂: multiset
U41₂: [1,2]
A_{_U41₂}: multiset
U51₃: [2,3,1]
A_{_U51₃}: [2,3,1]
A_{_PLUS₂}: [2,1]
0: multiset
and₂: multiset
U61₁: [1]
U71₃: [3,2,1]
A_{_U71₃}: [3,2,1]
a_{_x₂}: [1,2]
A_{_X₂}: [1,2]
a_{_U41₂}: [1,2]
a_{_plus₂}: [2,1]
a_{_U71₃}: [3,2,1]
a_{_U11₃}: multiset
a_{_U21₂}: multiset
a_{_U31₃}: multiset
a_{_U12₂}: multiset
a_{_U22₁}: [1]
a_{_U32₂}: multiset
a_{_U51₃}: [2,3,1]
a_{_U61₁}: [1]

The following usable rules [FROCOS05] were oriented:

a__isNatKind(0) → tt
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
a__U41(tt, N) → mark(N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isNatKind(X)) → a__isNatKind(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(X) → isNatKind(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(X) → isNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(X1, X2) → x(X1, X2)
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(tt) → tt
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U61(tt) → 0
a__U61(X) → U61(X)

(6) Obligation:

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

A__U12(tt, V2) → A__ISNAT(V2)
A__AND(tt, X) → MARK(X)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U13(X)) → MARK(X)
MARK(U33(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

MARK(U33(X)) → MARK(X)
MARK(U13(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U33(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
U33(x1) = U33(x1)
U13(x1) = x1

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

[MARK₁, U33₁]

Status:

MARK₁: [1]
U33₁: multiset

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

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

MARK(U13(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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.

MARK(U13(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

U13₁ > MARK₁

Status:

MARK₁: multiset
U13₁: multiset

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(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(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
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) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(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__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

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

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE