Termination proof of ../tpdb/TRS/CSR_Maude/my-nat/MYNAT

YES Termination proof of ../tpdb/TRS/CSR_Maude/my-nat/MYNAT_complete.trs
Termination of the following Term Rewriting System could be proven:

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

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U32: {1}
U33: {1}
U41: {1}
U51: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
x: {1, 2}
and: {1}
isNatKind: empty set

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

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

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

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

U11(tt, V1, V2) → U12(isNat(a(V1)), a(V2))
U12(tt, V2) → U13(isNat(a(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(a(V1)), a(V2))
U32(tt, V2) → U33(isNat(a(V2)))
U33(tt) → tt
U41(tt, N) → a(N)
U51(tt, M, N) → s(plus(a(N), a(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(a(N), a(M)), a(N))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(plusInact(V1, V2)) → U11(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNat(xInact(V1, V2)) → U31(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNatKind(0Inact) → tt
isNatKind(plusInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatKind(xInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
plus(N, 0) → U41(and(isNat(N), isNatKindInact(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKindInact(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
0 → 0Inact
a(0Inact) → 0
x(x1, x2) → xInact(x1, x2)
a(xInact(x1, x2)) → x(x1, x2)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
plus(x1, x2) → plusInact(x1, x2)
a(plusInact(x1, x2)) → plus(x1, x2)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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(isNatKindInact(x1)) → ISNATKIND(x1)
U31¹(tt, V1, V2) → ISNAT(a(V1))
U11¹(tt, V1, V2) → ISNAT(a(V1))
A(plusInact(x1, x2)) → PLUS(x1, x2)
ISNAT(plusInact(V1, V2)) → A(V1)
X(N, 0) → AND(isNat(N), isNatKindInact(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(xInact(V1, V2)) → U31¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
U11¹(tt, V1, V2) → U12¹(isNat(a(V1)), a(V2))
PLUS(N, s(M)) → ISNAT(M)
U61¹(tt) → 0¹
U12¹(tt, V2) → A(V2)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATKIND(xInact(V1, V2)) → A(V2)
PLUS(N, s(M)) → U51¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
U51¹(tt, M, N) → PLUS(a(N), a(M))
U21¹(tt, V1) → U22¹(isNat(a(V1)))
U71¹(tt, M, N) → A(N)
AND(tt, X) → A(X)
ISNAT(xInact(V1, V2)) → A(V2)
ISNAT(sInact(V1)) → U21¹(isNatKind(a(V1)), a(V1))
U71¹(tt, M, N) → A(M)
A(sInact(x1)) → S(x1)
U11¹(tt, V1, V2) → A(V1)
U21¹(tt, V1) → A(V1)
ISNAT(sInact(V1)) → A(V1)
X(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U51¹(tt, M, N) → A(N)
ISNATKIND(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
PLUS(N, 0) → AND(isNat(N), isNatKindInact(N))
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
PLUS(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U12¹(tt, V2) → U13¹(isNat(a(V2)))
U41¹(tt, N) → A(N)
ISNATKIND(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → A(V1)
ISNATKIND(plusInact(V1, V2)) → A(V1)
U32¹(tt, V2) → ISNAT(a(V2))
U71¹(tt, M, N) → X(a(N), a(M))
ISNAT(plusInact(V1, V2)) → U11¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
ISNATKIND(sInact(V1)) → A(V1)
PLUS(N, 0) → U41¹(and(isNat(N), isNatKindInact(N)), N)
PLUS(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
U32¹(tt, V2) → U33¹(isNat(a(V2)))
X(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(xInact(V1, V2)) → A(V1)
ISNAT(xInact(V1, V2)) → A(V1)
ISNATKIND(xInact(V1, V2)) → ISNATKIND(a(V1))
U21¹(tt, V1) → ISNAT(a(V1))
ISNAT(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → U32¹(isNat(a(V1)), a(V2))
ISNAT(xInact(V1, V2)) → ISNATKIND(a(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
A(0Inact) → 0¹
ISNAT(plusInact(V1, V2)) → A(V2)
ISNAT(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
U51¹(tt, M, N) → S(plus(a(N), a(M)))
A(xInact(x1, x2)) → X(x1, x2)
X(N, s(M)) → ISNAT(M)
X(N, 0) → U61¹(and(isNat(N), isNatKindInact(N)))
U12¹(tt, V2) → ISNAT(a(V2))
X(N, 0) → ISNAT(N)
U51¹(tt, M, N) → A(M)
ISNATKIND(plusInact(V1, V2)) → A(V2)
U11¹(tt, V1, V2) → A(V2)
U71¹(tt, M, N) → PLUS(x(a(N), a(M)), a(N))
ISNAT(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
X(N, s(M)) → ISNAT(N)
U31¹(tt, V1, V2) → A(V2)
PLUS(N, s(M)) → ISNAT(N)
U32¹(tt, V2) → A(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(a(V1)), a(V2))
U12(tt, V2) → U13(isNat(a(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(a(V1)), a(V2))
U32(tt, V2) → U33(isNat(a(V2)))
U33(tt) → tt
U41(tt, N) → a(N)
U51(tt, M, N) → s(plus(a(N), a(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(a(N), a(M)), a(N))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(plusInact(V1, V2)) → U11(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNat(xInact(V1, V2)) → U31(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNatKind(0Inact) → tt
isNatKind(plusInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatKind(xInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
plus(N, 0) → U41(and(isNat(N), isNatKindInact(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKindInact(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
0 → 0Inact
a(0Inact) → 0
x(x1, x2) → xInact(x1, x2)
a(xInact(x1, x2)) → x(x1, x2)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
plus(x1, x2) → plusInact(x1, x2)
a(plusInact(x1, x2)) → plus(x1, x2)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

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

A(isNatKindInact(x1)) → ISNATKIND(x1)
U31¹(tt, V1, V2) → ISNAT(a(V1))
U11¹(tt, V1, V2) → ISNAT(a(V1))
A(plusInact(x1, x2)) → PLUS(x1, x2)
ISNAT(plusInact(V1, V2)) → A(V1)
X(N, 0) → AND(isNat(N), isNatKindInact(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(xInact(V1, V2)) → U31¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
U11¹(tt, V1, V2) → U12¹(isNat(a(V1)), a(V2))
PLUS(N, s(M)) → ISNAT(M)
U61¹(tt) → 0¹
U12¹(tt, V2) → A(V2)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATKIND(xInact(V1, V2)) → A(V2)
PLUS(N, s(M)) → U51¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
U51¹(tt, M, N) → PLUS(a(N), a(M))
U21¹(tt, V1) → U22¹(isNat(a(V1)))
U71¹(tt, M, N) → A(N)
AND(tt, X) → A(X)
ISNAT(xInact(V1, V2)) → A(V2)
ISNAT(sInact(V1)) → U21¹(isNatKind(a(V1)), a(V1))
U71¹(tt, M, N) → A(M)
A(sInact(x1)) → S(x1)
U11¹(tt, V1, V2) → A(V1)
U21¹(tt, V1) → A(V1)
ISNAT(sInact(V1)) → A(V1)
X(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U51¹(tt, M, N) → A(N)
ISNATKIND(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
PLUS(N, 0) → AND(isNat(N), isNatKindInact(N))
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
PLUS(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U12¹(tt, V2) → U13¹(isNat(a(V2)))
U41¹(tt, N) → A(N)
ISNATKIND(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → A(V1)
ISNATKIND(plusInact(V1, V2)) → A(V1)
U32¹(tt, V2) → ISNAT(a(V2))
U71¹(tt, M, N) → X(a(N), a(M))
ISNAT(plusInact(V1, V2)) → U11¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
ISNATKIND(sInact(V1)) → A(V1)
PLUS(N, 0) → U41¹(and(isNat(N), isNatKindInact(N)), N)
PLUS(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
U32¹(tt, V2) → U33¹(isNat(a(V2)))
X(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(xInact(V1, V2)) → A(V1)
ISNAT(xInact(V1, V2)) → A(V1)
ISNATKIND(xInact(V1, V2)) → ISNATKIND(a(V1))
U21¹(tt, V1) → ISNAT(a(V1))
ISNAT(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → U32¹(isNat(a(V1)), a(V2))
ISNAT(xInact(V1, V2)) → ISNATKIND(a(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
A(0Inact) → 0¹
ISNAT(plusInact(V1, V2)) → A(V2)
ISNAT(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
U51¹(tt, M, N) → S(plus(a(N), a(M)))
A(xInact(x1, x2)) → X(x1, x2)
X(N, s(M)) → ISNAT(M)
X(N, 0) → U61¹(and(isNat(N), isNatKindInact(N)))
U12¹(tt, V2) → ISNAT(a(V2))
X(N, 0) → ISNAT(N)
U51¹(tt, M, N) → A(M)
ISNATKIND(plusInact(V1, V2)) → A(V2)
U11¹(tt, V1, V2) → A(V2)
U71¹(tt, M, N) → PLUS(x(a(N), a(M)), a(N))
ISNAT(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
X(N, s(M)) → ISNAT(N)
U31¹(tt, V1, V2) → A(V2)
PLUS(N, s(M)) → ISNAT(N)
U32¹(tt, V2) → A(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(a(V1)), a(V2))
U12(tt, V2) → U13(isNat(a(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(a(V1)), a(V2))
U32(tt, V2) → U33(isNat(a(V2)))
U33(tt) → tt
U41(tt, N) → a(N)
U51(tt, M, N) → s(plus(a(N), a(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(a(N), a(M)), a(N))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(plusInact(V1, V2)) → U11(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNat(xInact(V1, V2)) → U31(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNatKind(0Inact) → tt
isNatKind(plusInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatKind(xInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
plus(N, 0) → U41(and(isNat(N), isNatKindInact(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKindInact(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
0 → 0Inact
a(0Inact) → 0
x(x1, x2) → xInact(x1, x2)
a(xInact(x1, x2)) → x(x1, x2)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
plus(x1, x2) → plusInact(x1, x2)
a(plusInact(x1, x2)) → plus(x1, x2)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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 8 less nodes.

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ QDPOrderProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

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

A(isNatKindInact(x1)) → ISNATKIND(x1)
U31¹(tt, V1, V2) → ISNAT(a(V1))
U11¹(tt, V1, V2) → ISNAT(a(V1))
ISNAT(plusInact(V1, V2)) → A(V1)
A(plusInact(x1, x2)) → PLUS(x1, x2)
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), isNatKindInact(N))
ISNAT(xInact(V1, V2)) → U31¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
U11¹(tt, V1, V2) → U12¹(isNat(a(V1)), a(V2))
PLUS(N, s(M)) → ISNAT(M)
U12¹(tt, V2) → A(V2)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATKIND(xInact(V1, V2)) → A(V2)
PLUS(N, s(M)) → U51¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
U51¹(tt, M, N) → PLUS(a(N), a(M))
U71¹(tt, M, N) → A(N)
AND(tt, X) → A(X)
ISNAT(xInact(V1, V2)) → A(V2)
ISNAT(sInact(V1)) → U21¹(isNatKind(a(V1)), a(V1))
U71¹(tt, M, N) → A(M)
U11¹(tt, V1, V2) → A(V1)
ISNAT(sInact(V1)) → A(V1)
U21¹(tt, V1) → A(V1)
X(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U51¹(tt, M, N) → A(N)
ISNATKIND(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
PLUS(N, 0) → AND(isNat(N), isNatKindInact(N))
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
PLUS(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U41¹(tt, N) → A(N)
ISNATKIND(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → A(V1)
ISNATKIND(plusInact(V1, V2)) → A(V1)
U32¹(tt, V2) → ISNAT(a(V2))
U71¹(tt, M, N) → X(a(N), a(M))
ISNAT(plusInact(V1, V2)) → U11¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
ISNATKIND(sInact(V1)) → A(V1)
PLUS(N, 0) → U41¹(and(isNat(N), isNatKindInact(N)), N)
PLUS(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
X(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(xInact(V1, V2)) → A(V1)
ISNAT(xInact(V1, V2)) → A(V1)
ISNATKIND(xInact(V1, V2)) → ISNATKIND(a(V1))
U21¹(tt, V1) → ISNAT(a(V1))
ISNAT(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → U32¹(isNat(a(V1)), a(V2))
ISNAT(xInact(V1, V2)) → ISNATKIND(a(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
ISNAT(plusInact(V1, V2)) → A(V2)
ISNAT(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
A(xInact(x1, x2)) → X(x1, x2)
X(N, s(M)) → ISNAT(M)
U12¹(tt, V2) → ISNAT(a(V2))
X(N, 0) → ISNAT(N)
U51¹(tt, M, N) → A(M)
ISNATKIND(plusInact(V1, V2)) → A(V2)
U11¹(tt, V1, V2) → A(V2)
U71¹(tt, M, N) → PLUS(x(a(N), a(M)), a(N))
ISNAT(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
X(N, s(M)) → ISNAT(N)
U31¹(tt, V1, V2) → A(V2)
PLUS(N, s(M)) → ISNAT(N)
U32¹(tt, V2) → A(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(a(V1)), a(V2))
U12(tt, V2) → U13(isNat(a(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(a(V1)), a(V2))
U32(tt, V2) → U33(isNat(a(V2)))
U33(tt) → tt
U41(tt, N) → a(N)
U51(tt, M, N) → s(plus(a(N), a(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(a(N), a(M)), a(N))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(plusInact(V1, V2)) → U11(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNat(xInact(V1, V2)) → U31(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNatKind(0Inact) → tt
isNatKind(plusInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatKind(xInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
plus(N, 0) → U41(and(isNat(N), isNatKindInact(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKindInact(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
0 → 0Inact
a(0Inact) → 0
x(x1, x2) → xInact(x1, x2)
a(xInact(x1, x2)) → x(x1, x2)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
plus(x1, x2) → plusInact(x1, x2)
a(plusInact(x1, x2)) → plus(x1, x2)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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(isNatKindInact(x1)) → ISNATKIND(x1)
U31¹(tt, V1, V2) → ISNAT(a(V1))
U11¹(tt, V1, V2) → ISNAT(a(V1))
ISNAT(plusInact(V1, V2)) → A(V1)
A(plusInact(x1, x2)) → PLUS(x1, x2)
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), isNatKindInact(N))
ISNAT(xInact(V1, V2)) → U31¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
U11¹(tt, V1, V2) → U12¹(isNat(a(V1)), a(V2))
PLUS(N, s(M)) → ISNAT(M)
U12¹(tt, V2) → A(V2)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATKIND(xInact(V1, V2)) → A(V2)
PLUS(N, s(M)) → U51¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
U71¹(tt, M, N) → A(N)
ISNAT(xInact(V1, V2)) → A(V2)
ISNAT(sInact(V1)) → U21¹(isNatKind(a(V1)), a(V1))
U71¹(tt, M, N) → A(M)
U11¹(tt, V1, V2) → A(V1)
ISNAT(sInact(V1)) → A(V1)
X(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U51¹(tt, M, N) → A(N)
ISNATKIND(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
PLUS(N, 0) → AND(isNat(N), isNatKindInact(N))
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
PLUS(N, s(M)) → AND(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N)))
U41¹(tt, N) → A(N)
ISNATKIND(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → A(V1)
ISNATKIND(plusInact(V1, V2)) → A(V1)
U32¹(tt, V2) → ISNAT(a(V2))
U71¹(tt, M, N) → X(a(N), a(M))
ISNAT(plusInact(V1, V2)) → U11¹(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
ISNATKIND(sInact(V1)) → A(V1)
PLUS(N, 0) → U41¹(and(isNat(N), isNatKindInact(N)), N)
PLUS(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
X(N, s(M)) → AND(isNat(M), isNatKindInact(M))
ISNATKIND(xInact(V1, V2)) → A(V1)
ISNAT(xInact(V1, V2)) → A(V1)
ISNATKIND(xInact(V1, V2)) → ISNATKIND(a(V1))
ISNAT(plusInact(V1, V2)) → ISNATKIND(a(V1))
U31¹(tt, V1, V2) → U32¹(isNat(a(V1)), a(V2))
ISNAT(xInact(V1, V2)) → ISNATKIND(a(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
ISNAT(plusInact(V1, V2)) → A(V2)
ISNAT(plusInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
A(xInact(x1, x2)) → X(x1, x2)
X(N, s(M)) → ISNAT(M)
U12¹(tt, V2) → ISNAT(a(V2))
X(N, 0) → ISNAT(N)
U51¹(tt, M, N) → A(M)
ISNATKIND(plusInact(V1, V2)) → A(V2)
U11¹(tt, V1, V2) → A(V2)
U71¹(tt, M, N) → PLUS(x(a(N), a(M)), a(N))
ISNAT(xInact(V1, V2)) → AND(isNatKind(a(V1)), isNatKindInact(a(V2)))
X(N, s(M)) → ISNAT(N)
U31¹(tt, V1, V2) → A(V2)
PLUS(N, s(M)) → ISNAT(N)
U32¹(tt, V2) → A(V2)

The remaining pairs can at least be oriented weakly.

U51¹(tt, M, N) → PLUS(a(N), a(M))
AND(tt, X) → A(X)
U21¹(tt, V1) → A(V1)
U21¹(tt, V1) → ISNAT(a(V1))

Used ordering: Combined order from the following AFS and order.
A(x1) = A(x1)
isNatKindInact(x1) = isNatKindInact(x1)
ISNATKIND(x1) = ISNATKIND(x1)
U31¹(x1, x2, x3) = U31¹(x1, x2, x3)
tt = tt
ISNAT(x1) = ISNAT(x1)
a(x1) = x1
U11¹(x1, x2, x3) = U11¹(x1, x2, x3)
plusInact(x1, x2) = plusInact(x1, x2)
PLUS(x1, x2) = PLUS(x1, x2)
0 = 0
X(x1, x2) = X(x1, x2)
AND(x1, x2) = AND(x2)
isNat(x1) = isNat
xInact(x1, x2) = xInact(x1, x2)
and(x1, x2) = and(x2)
isNatKind(x1) = isNatKind(x1)
U12¹(x1, x2) = U12¹(x2)
s(x1) = s(x1)
andInact(x1, x2) = andInact(x2)
U51¹(x1, x2, x3) = U51¹(x2, x3)
U71¹(x1, x2, x3) = U71¹(x1, x2, x3)
sInact(x1) = sInact(x1)
U21¹(x1, x2) = U21¹(x2)
U41¹(x1, x2) = U41¹(x2)
U32¹(x1, x2) = U32¹(x2)
x(x1, x2) = x(x1, x2)
plus(x1, x2) = plus(x1, x2)
U41(x1, x2) = U41(x1, x2)
U71(x1, x2, x3) = U71(x1, x2, x3)
0Inact = 0Inact
U11(x1, x2, x3) = U11
U12(x1, x2) = x1
U21(x1, x2) = U21
U51(x1, x2, x3) = U51(x1, x2, x3)
U61(x1) = U61
U31(x1, x2, x3) = U31
U32(x1, x2) = U32
U22(x1) = x1
U33(x1) = U33
U13(x1) = U13

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

[X₂, xInact₂, U71^1₃, x₂, U71_3] > U31^1₃ > U32^1₁ > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [isNatKindInact₁, isNatKind_1] > [tt, U13]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > U31^1₃ > U32^1₁ > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [and₁, andInact_1]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [plusInact₂, plus₂, U51_3] > U11^1₃ > U12^1₁ > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [isNatKindInact₁, isNatKind_1] > [tt, U13]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [plusInact₂, plus₂, U51_3] > U11^1₃ > U12^1₁ > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [and₁, andInact_1]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [plusInact₂, plus₂, U51_3] > [PLUS₂, U51^1₂, U41^1_1] > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [isNatKindInact₁, isNatKind_1] > [tt, U13]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [plusInact₂, plus₂, U51_3] > [PLUS₂, U51^1₂, U41^1_1] > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [and₁, andInact_1]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [plusInact₂, plus₂, U51_3] > U41₂
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [isNat, U11, U21] > [isNatKindInact₁, isNatKind_1] > [tt, U13]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [isNat, U11, U21] > [and₁, andInact_1]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > [isNat, U11, U21] > U31 > [U32, U33] > [tt, U13]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > U61 > [0, 0Inact] > [PLUS₂, U51^1₂, U41^1_1] > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [isNatKindInact₁, isNatKind_1] > [tt, U13]
[X₂, xInact₂, U71^1₃, x₂, U71_3] > U61 > [0, 0Inact] > [PLUS₂, U51^1₂, U41^1_1] > [A₁, ISNATKIND₁, ISNAT₁, AND₁, s₁, sInact₁, U21^1_1] > [and₁, andInact_1]

Status:

isNatKind₁: [1]
U21^1₁: [1]
U33: []
U21: []
0: multiset
0Inact: multiset
U11: []
U31^1₃: multiset
ISNAT₁: [1]
U71^1₃: [3,2,1]
U31: multiset
ISNATKIND₁: [1]
U13: multiset
U41^1₁: [1]
X₂: [1,2]
s₁: multiset
isNatKindInact₁: [1]
xInact₂: [1,2]
U71₃: [3,2,1]
isNat: []
U11^1₃: multiset
U41₂: multiset
U32: []
A₁: [1]
PLUS₂: [1,2]
x₂: [1,2]
tt: multiset
U61: multiset
AND₁: [1]
U51^1₂: [2,1]
plus₂: [2,1]
andInact₁: multiset
sInact₁: multiset
U51₃: [2,3,1]
U12^1₁: [1]
and₁: multiset
U32^1₁: [1]
plusInact₂: [2,1]

The following usable rules [17] were oriented:

isNatKind(plusInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
plus(N, 0) → U41(and(isNat(N), isNatKindInact(N)), N)
isNatKind(xInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
and(tt, X) → a(X)
U41(tt, N) → a(N)
a(xInact(x1, x2)) → x(x1, x2)
isNatKind(sInact(V1)) → isNatKind(a(V1))
a(andInact(x1, x2)) → and(x1, x2)
x(N, s(M)) → U71(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
U71(tt, M, N) → plus(x(a(N), a(M)), a(N))
a(plusInact(x1, x2)) → plus(x1, x2)
a(isNatKindInact(x1)) → isNatKind(x1)
0 → 0Inact
isNat(0Inact) → tt
a(0Inact) → 0
isNatKind(x1) → isNatKindInact(x1)
U11(tt, V1, V2) → U12(isNat(a(V1)), a(V2))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNat(plusInact(V1, V2)) → U11(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
U51(tt, M, N) → s(plus(a(N), a(M)))
U61(tt) → 0
a(x) → x
plus(x1, x2) → plusInact(x1, x2)
and(x1, x2) → andInact(x1, x2)
U31(tt, V1, V2) → U32(isNat(a(V1)), a(V2))
a(sInact(x1)) → s(x1)
U22(tt) → tt
x(x1, x2) → xInact(x1, x2)
U21(tt, V1) → U22(isNat(a(V1)))
U32(tt, V2) → U33(isNat(a(V2)))
U12(tt, V2) → U13(isNat(a(V2)))
isNat(xInact(V1, V2)) → U31(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
x(N, 0) → U61(and(isNat(N), isNatKindInact(N)))
s(x1) → sInact(x1)
isNatKind(0Inact) → tt
U13(tt) → tt
plus(N, s(M)) → U51(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
U33(tt) → tt

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

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

U21¹(tt, V1) → A(V1)
U21¹(tt, V1) → ISNAT(a(V1))
U51¹(tt, M, N) → PLUS(a(N), a(M))
AND(tt, X) → A(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(a(V1)), a(V2))
U12(tt, V2) → U13(isNat(a(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(a(V1)), a(V2))
U32(tt, V2) → U33(isNat(a(V2)))
U33(tt) → tt
U41(tt, N) → a(N)
U51(tt, M, N) → s(plus(a(N), a(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(a(N), a(M)), a(N))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(plusInact(V1, V2)) → U11(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNat(xInact(V1, V2)) → U31(and(isNatKind(a(V1)), isNatKindInact(a(V2))), a(V1), a(V2))
isNatKind(0Inact) → tt
isNatKind(plusInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatKind(xInact(V1, V2)) → and(isNatKind(a(V1)), isNatKindInact(a(V2)))
plus(N, 0) → U41(and(isNat(N), isNatKindInact(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKindInact(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKindInact(M)), andInact(isNat(N), isNatKindInact(N))), M, N)
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
0 → 0Inact
a(0Inact) → 0
x(x1, x2) → xInact(x1, x2)
a(xInact(x1, x2)) → x(x1, x2)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
plus(x1, x2) → plusInact(x1, x2)
a(plusInact(x1, x2)) → plus(x1, x2)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Incomplete Giesl Middeldorp [11] to transform the context-sensitive TRS to an usual TRS.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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:

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U33(x1)) → U33ACTIVE(mark(x1))
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U11ACTIVE(tt, V1, V2) → U12ACTIVE(isNatActive(V1), V2)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U61(x1)) → U61ACTIVE(mark(x1))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
MARK(U22(x1)) → U22ACTIVE(mark(x1))
U12ACTIVE(tt, V2) → U13ACTIVE(isNatActive(V2))
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(s(x1)) → MARK(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → U61ACTIVE(andActive(isNatActive(N), isNatKind(N)))
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U33(x1)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U13(x1)) → U13ACTIVE(mark(x1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U32ACTIVE(tt, V2) → U33ACTIVE(isNatActive(V2))
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U21ACTIVE(tt, V1) → U22ACTIVE(isNatActive(V1))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U33(x1)) → U33ACTIVE(mark(x1))
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U11ACTIVE(tt, V1, V2) → U12ACTIVE(isNatActive(V1), V2)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U61(x1)) → U61ACTIVE(mark(x1))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
MARK(U22(x1)) → U22ACTIVE(mark(x1))
U12ACTIVE(tt, V2) → U13ACTIVE(isNatActive(V2))
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(s(x1)) → MARK(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → U61ACTIVE(andActive(isNatActive(N), isNatKind(N)))
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U33(x1)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U13(x1)) → U13ACTIVE(mark(x1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U32ACTIVE(tt, V2) → U33ACTIVE(isNatActive(V2))
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U21ACTIVE(tt, V1) → U22ACTIVE(isNatActive(V1))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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 8 less nodes.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U11ACTIVE(tt, V1, V2) → U12ACTIVE(isNatActive(V1), V2)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(x1)) → MARK(x1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U11ACTIVE(tt, V1, V2) → U12ACTIVE(isNatActive(V1), V2) at position [0] we obtained the following new rules:

U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
U11ACTIVE(tt, x0, y1) → U12ACTIVE(isNat(x0), y1)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
MARK(U21(x1, x2)) → MARK(x1)
ANDACTIVE(tt, X) → MARK(X)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(s(x1)) → MARK(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U33(x1)) → MARK(x1)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
U11ACTIVE(tt, x0, y1) → U12ACTIVE(isNat(x0), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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 1 less node.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

ISNATACTIVE(plus(x0, y1)) → ANDACTIVE(isNatKind(x0), isNatKind(y1))
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U12(x1, x2)) → MARK(x1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
MARK(U21(x1, x2)) → MARK(x1)
ANDACTIVE(tt, X) → MARK(X)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(s(x1)) → MARK(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATACTIVE(plus(x0, y1)) → ANDACTIVE(isNatKind(x0), isNatKind(y1))
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U33(x1)) → MARK(x1)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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 1 less node.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATKINDACTIVE(plus(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATKINDACTIVE(plus(x0, y1)) → ANDACTIVE(isNatKind(x0), isNatKind(y1))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
MARK(U21(x1, x2)) → MARK(x1)
ANDACTIVE(tt, X) → MARK(X)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(s(x1)) → MARK(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
ISNATKINDACTIVE(plus(x0, y1)) → ANDACTIVE(isNatKind(x0), isNatKind(y1))
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U33(x1)) → MARK(x1)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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 1 less node.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U33(x1)) → MARK(x1)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATKINDACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

ISNATKINDACTIVE(x(x0, y1)) → ANDACTIVE(isNatKind(x0), isNatKind(y1))
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U12(x1, x2)) → MARK(x1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(x1, x2)) → MARK(x1)
ANDACTIVE(tt, X) → MARK(X)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
ISNATKINDACTIVE(x(x0, y1)) → ANDACTIVE(isNatKind(x0), isNatKind(y1))
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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 1 less node.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
U51ACTIVE(tt, M, N) → MARK(M)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
XACTIVE(N, 0) → ISNATACTIVE(N)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(x1, x2)) → MARK(x1)
U51ACTIVE(tt, M, N) → MARK(N)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U51(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(plus(x1, x2)) → MARK(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U33(x1)) → MARK(x1)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U11(x1, x2, x3)) → U11ACTIVE(mark(x1), x2, x3) at position [0] we obtained the following new rules:

MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(0, y1, y2)) → U11ACTIVE(0, y1, y2)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U11(s(x0), y1, y2)) → U11ACTIVE(s(mark(x0)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(s(x0), y1, y2)) → U11ACTIVE(s(mark(x0)), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
MARK(U11(0, y1, y2)) → U11ACTIVE(0, y1, y2)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U12(x1, x2)) → U12ACTIVE(mark(x1), x2) at position [0] we obtained the following new rules:

MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U12(s(x0), y1)) → U12ACTIVE(s(mark(x0)), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(0, y1)) → U12ACTIVE(0, y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U12(s(x0), y1)) → U12ACTIVE(s(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U12(0, y1)) → U12ACTIVE(0, y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(x1, x2)) → MARK(x1)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(x1, x2)) → MARK(x1)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2) at position [0] we obtained the following new rules:

MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(U21(s(x0), y1)) → U21ACTIVE(s(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(0, y1)) → U21ACTIVE(0, y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
MARK(U21(s(x0), y1)) → U21ACTIVE(s(mark(x0)), y1)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U21(0, y1)) → U21ACTIVE(0, y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(x1, x2)) → MARK(x1)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U31(x1, x2, x3)) → U31ACTIVE(mark(x1), x2, x3) at position [0] we obtained the following new rules:

MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U31(0, y1, y2)) → U31ACTIVE(0, y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U31(s(x0), y1, y2)) → U31ACTIVE(s(mark(x0)), y1, y2)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U31(0, y1, y2)) → U31ACTIVE(0, y1, y2)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U31(s(x0), y1, y2)) → U31ACTIVE(s(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U32(x1, x2)) → U32ACTIVE(mark(x1), x2) at position [0] we obtained the following new rules:

MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U32(s(x0), y1)) → U32ACTIVE(s(mark(x0)), y1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U32(0, y1)) → U32ACTIVE(0, y1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U32(0, y1)) → U32ACTIVE(0, y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(s(x0), y1)) → U32ACTIVE(s(mark(x0)), y1)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2)
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U41(x1, x2)) → U41ACTIVE(mark(x1), x2) at position [0] we obtained the following new rules:

MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U41(0, y1)) → U41ACTIVE(0, y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U41(s(x0), y1)) → U41ACTIVE(s(mark(x0)), y1)
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U41(0, y1)) → U41ACTIVE(0, y1)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(s(x0), y1)) → U41ACTIVE(s(mark(x0)), y1)
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(plus(x1, x2)) → MARK(x1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U22(x1)) → MARK(x1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3) at position [0] we obtained the following new rules:

MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(s(x0), y1, y2)) → U51ACTIVE(s(mark(x0)), y1, y2)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U51(0, y1, y2)) → U51ACTIVE(0, y1, y2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → MARK(x1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U22(x1)) → MARK(x1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U51(0, y1, y2)) → U51ACTIVE(0, y1, y2)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U51(s(x0), y1, y2)) → U51ACTIVE(s(mark(x0)), y1, y2)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → MARK(x1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U71(x1, x2, x3)) → U71ACTIVE(mark(x1), x2, x3) at position [0] we obtained the following new rules:

MARK(U71(U32(x0, x1), y1, y2)) → U71ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → U71ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U71(U71(x0, x1, x2), y1, y2)) → U71ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U71(s(x0), y1, y2)) → U71ACTIVE(s(mark(x0)), y1, y2)
MARK(U71(U22(x0), y1, y2)) → U71ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → U71ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U71(tt, y1, y2)) → U71ACTIVE(tt, y1, y2)
MARK(U71(U11(x0, x1, x2), y1, y2)) → U71ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U71(U33(x0), y1, y2)) → U71ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U71(U61(x0), y1, y2)) → U71ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → U71ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → U71ACTIVE(isNatActive(x0), y1, y2)
MARK(U71(U31(x0, x1, x2), y1, y2)) → U71ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U71(0, y1, y2)) → U71ACTIVE(0, y1, y2)
MARK(U71(and(x0, x1), y1, y2)) → U71ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U71(U13(x0), y1, y2)) → U71ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U71(U12(x0, x1), y1, y2)) → U71ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(plus(x0, x1), y1, y2)) → U71ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(x(x0, x1), y1, y2)) → U71ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(U51(x0, x1, x2), y1, y2)) → U71ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → U71ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U71(0, y1, y2)) → U71ACTIVE(0, y1, y2)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → MARK(x1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
MARK(U71(s(x0), y1, y2)) → U71ACTIVE(s(mark(x0)), y1, y2)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U22(x1)) → MARK(x1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U71(x0, x1, x2), y1, y2)) → U71ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(U71(U32(x0, x1), y1, y2)) → U71ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U71(U13(x0), y1, y2)) → U71ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U71(U61(x0), y1, y2)) → U71ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U71(tt, y1, y2)) → U71ACTIVE(tt, y1, y2)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U71(x(x0, x1), y1, y2)) → U71ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U71(U41(x0, x1), y1, y2)) → U71ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
MARK(U71(and(x0, x1), y1, y2)) → U71ACTIVE(andActive(mark(x0), x1), y1, y2)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U71(isNatKind(x0), y1, y2)) → U71ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U71(U33(x0), y1, y2)) → U71ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U71(U51(x0, x1, x2), y1, y2)) → U71ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U71(U11(x0, x1, x2), y1, y2)) → U71ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → U71ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U71(plus(x0, x1), y1, y2)) → U71ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(isNat(x0), y1, y2)) → U71ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U71(U22(x0), y1, y2)) → U71ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U71(U12(x0, x1), y1, y2)) → U71ACTIVE(U12Active(mark(x0), x1), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → U71ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → MARK(x1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U22(x1)) → MARK(x1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U71(x0, x1, x2), y1, y2)) → U71ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U12(x1, x2)) → MARK(x1)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(U71(U32(x0, x1), y1, y2)) → U71ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U71(U13(x0), y1, y2)) → U71ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U71(U61(x0), y1, y2)) → U71ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U71(tt, y1, y2)) → U71ACTIVE(tt, y1, y2)
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U71(x(x0, x1), y1, y2)) → U71ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U71(U41(x0, x1), y1, y2)) → U71ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U71(and(x0, x1), y1, y2)) → U71ACTIVE(andActive(mark(x0), x1), y1, y2)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
U51ACTIVE(tt, M, N) → MARK(N)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → U71ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U71(U33(x0), y1, y2)) → U71ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U71(U51(x0, x1, x2), y1, y2)) → U71ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U61(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U71(U11(x0, x1, x2), y1, y2)) → U71ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → U71ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U71(plus(x0, x1), y1, y2)) → U71ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(isNat(x0), y1, y2)) → U71ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U71(U22(x0), y1, y2)) → U71ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U71(U12(x0, x1), y1, y2)) → U71ACTIVE(U12Active(mark(x0), x1), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2) at position [0] we obtained the following new rules:

MARK(and(0, y1)) → ANDACTIVE(0, y1)
MARK(and(U61(x0), y1)) → ANDACTIVE(U61Active(mark(x0)), y1)
MARK(and(isNat(x0), y1)) → ANDACTIVE(isNatActive(x0), y1)
MARK(and(U12(x0, x1), y1)) → ANDACTIVE(U12Active(mark(x0), x1), y1)
MARK(and(plus(x0, x1), y1)) → ANDACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → ANDACTIVE(isNatKindActive(x0), y1)
MARK(and(U71(x0, x1, x2), y1)) → ANDACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(and(U13(x0), y1)) → ANDACTIVE(U13Active(mark(x0)), y1)
MARK(and(U22(x0), y1)) → ANDACTIVE(U22Active(mark(x0)), y1)
MARK(and(and(x0, x1), y1)) → ANDACTIVE(andActive(mark(x0), x1), y1)
MARK(and(U11(x0, x1, x2), y1)) → ANDACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → ANDACTIVE(U33Active(mark(x0)), y1)
MARK(and(tt, y1)) → ANDACTIVE(tt, y1)
MARK(and(U21(x0, x1), y1)) → ANDACTIVE(U21Active(mark(x0), x1), y1)
MARK(and(x(x0, x1), y1)) → ANDACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(U51(x0, x1, x2), y1)) → ANDACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(and(s(x0), y1)) → ANDACTIVE(s(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → ANDACTIVE(U32Active(mark(x0), x1), y1)
MARK(and(U31(x0, x1, x2), y1)) → ANDACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(and(U41(x0, x1), y1)) → ANDACTIVE(U41Active(mark(x0), x1), y1)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(and(0, y1)) → ANDACTIVE(0, y1)
U51ACTIVE(tt, M, N) → MARK(M)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → U71ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(and(U41(x0, x1), y1)) → ANDACTIVE(U41Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(U11(x0, x1, x2), y1)) → ANDACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → MARK(x1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(and(U71(x0, x1, x2), y1)) → ANDACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U22(x1)) → MARK(x1)
MARK(and(U13(x0), y1)) → ANDACTIVE(U13Active(mark(x0)), y1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(and(tt, y1)) → ANDACTIVE(tt, y1)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U71(x0, x1, x2), y1, y2)) → U71ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U71(U32(x0, x1), y1, y2)) → U71ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U71(U13(x0), y1, y2)) → U71ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(and(isNat(x0), y1)) → ANDACTIVE(isNatActive(x0), y1)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U71(tt, y1, y2)) → U71ACTIVE(tt, y1, y2)
MARK(U71(U61(x0), y1, y2)) → U71ACTIVE(U61Active(mark(x0)), y1, y2)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U71(x(x0, x1), y1, y2)) → U71ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U71(U41(x0, x1), y1, y2)) → U71ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(and(U22(x0), y1)) → ANDACTIVE(U22Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(and(U51(x0, x1, x2), y1)) → ANDACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U32(x1, x2)) → MARK(x1)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(x(x1, x2)) → MARK(x2)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
MARK(U71(and(x0, x1), y1, y2)) → U71ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(and(U21(x0, x1), y1)) → ANDACTIVE(U21Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(and(x(x0, x1), y1)) → ANDACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(U32(x0, x1), y1)) → ANDACTIVE(U32Active(mark(x0), x1), y1)
MARK(and(s(x0), y1)) → ANDACTIVE(s(mark(x0)), y1)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U71(isNatKind(x0), y1, y2)) → U71ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(s(x1)) → MARK(x1)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(and(U31(x0, x1, x2), y1)) → ANDACTIVE(U31Active(mark(x0), x1, x2), y1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(and(U61(x0), y1)) → ANDACTIVE(U61Active(mark(x0)), y1)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U71(U33(x0), y1, y2)) → U71ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(and(and(x0, x1), y1)) → ANDACTIVE(andActive(mark(x0), x1), y1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U71(U51(x0, x1, x2), y1, y2)) → U71ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → ANDACTIVE(U12Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U71(U11(x0, x1, x2), y1, y2)) → U71ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(and(U33(x0), y1)) → ANDACTIVE(U33Active(mark(x0)), y1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(and(plus(x0, x1), y1)) → ANDACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → U71ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(and(isNatKind(x0), y1)) → ANDACTIVE(isNatKindActive(x0), y1)
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U71(plus(x0, x1), y1, y2)) → U71ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(isNat(x0), y1, y2)) → U71ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U71(U22(x0), y1, y2)) → U71ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U71(U12(x0, x1), y1, y2)) → U71ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1, x2, x3)) → U11Active(mark(x1), x2, x3)
U11Active(x1, x2, x3) → U11(x1, x2, x3)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(U13(x1)) → U13Active(mark(x1))
U13Active(x1) → U13(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2, x3)) → U31Active(mark(x1), x2, x3)
U31Active(x1, x2, x3) → U31(x1, x2, x3)
mark(U32(x1, x2)) → U32Active(mark(x1), x2)
U32Active(x1, x2) → U32(x1, x2)
mark(U33(x1)) → U33Active(mark(x1))
U33Active(x1) → U33(x1)
mark(U41(x1, x2)) → U41Active(mark(x1), x2)
U41Active(x1, x2) → U41(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U61(x1)) → U61Active(mark(x1))
U61Active(x1) → U61(x1)
mark(U71(x1, x2, x3)) → U71Active(mark(x1), x2, x3)
U71Active(x1, x2, x3) → U71(x1, x2, x3)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(plus(x1, x2)) → plusActive(mark(x1), mark(x2))
plusActive(x1, x2) → plus(x1, x2)
mark(x(x1, x2)) → xActive(mark(x1), mark(x2))
xActive(x1, x2) → x(x1, x2)
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(0) → 0
U11Active(tt, V1, V2) → U12Active(isNatActive(V1), V2)
U12Active(tt, V2) → U13Active(isNatActive(V2))
U13Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V1, V2) → U32Active(isNatActive(V1), V2)
U32Active(tt, V2) → U33Active(isNatActive(V2))
U33Active(tt) → tt
U41Active(tt, N) → mark(N)
U51Active(tt, M, N) → s(plusActive(mark(N), mark(M)))
U61Active(tt) → 0
U71Active(tt, M, N) → plusActive(xActive(mark(N), mark(M)), mark(N))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(plus(V1, V2)) → U11Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatActive(x(V1, V2)) → U31Active(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
isNatKindActive(0) → tt
isNatKindActive(plus(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatKindActive(x(V1, V2)) → andActive(isNatKindActive(V1), isNatKind(V2))
plusActive(N, 0) → U41Active(andActive(isNatActive(N), isNatKind(N)), N)
plusActive(N, s(M)) → U51Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
xActive(N, 0) → U61Active(andActive(isNatActive(N), isNatKind(N)))
xActive(N, s(M)) → U71Active(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

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

U51ACTIVE(tt, M, N) → MARK(M)
MARK(U11(U31(x0, x1, x2), y1, y2)) → U11ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U21(isNatKind(x0), y1)) → U21ACTIVE(isNatKindActive(x0), y1)
MARK(U32(isNat(x0), y1)) → U32ACTIVE(isNatActive(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → U71ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
ISNATACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
ISNATACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
ANDACTIVE(tt, X) → MARK(X)
ISNATKINDACTIVE(x(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U71(x0, x1, x2), y1)) → U12ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U51(U31(x0, x1, x2), y1, y2)) → U51ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → U11ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U12(U22(x0), y1)) → U12ACTIVE(U22Active(mark(x0)), y1)
MARK(and(U41(x0, x1), y1)) → ANDACTIVE(U41Active(mark(x0), x1), y1)
MARK(U21(U32(x0, x1), y1)) → U21ACTIVE(U32Active(mark(x0), x1), y1)
MARK(U12(isNatKind(x0), y1)) → U12ACTIVE(isNatKindActive(x0), y1)
MARK(U51(U71(x0, x1, x2), y1, y2)) → U51ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → U31ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U21(x(x0, x1), y1)) → U21ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U31(U41(x0, x1), y1, y2)) → U31ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(U11(x0, x1, x2), y1)) → ANDACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → U12ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → U21ACTIVE(U51Active(mark(x0), x1, x2), y1)
ISNATACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → U51ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(plus(x1, x2)) → MARK(x1)
MARK(U21(and(x0, x1), y1)) → U21ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U12(x0, x1), y1)) → U21ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → U41ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(tt, y1)) → U12ACTIVE(tt, y1)
MARK(U11(U33(x0), y1, y2)) → U11ACTIVE(U33Active(mark(x0)), y1, y2)
U31ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(and(U71(x0, x1, x2), y1)) → ANDACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U12(U41(x0, x1), y1)) → U12ACTIVE(U41Active(mark(x0), x1), y1)
ISNATACTIVE(plus(V1, V2)) → U11ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U11(U41(x0, x1), y1, y2)) → U11ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(and(x1, x2)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U41(isNatKind(x0), y1)) → U41ACTIVE(isNatKindActive(x0), y1)
MARK(U22(x1)) → MARK(x1)
MARK(and(U13(x0), y1)) → ANDACTIVE(U13Active(mark(x0)), y1)
U11ACTIVE(tt, 0, y1) → U12ACTIVE(tt, y1)
MARK(U41(U12(x0, x1), y1)) → U41ACTIVE(U12Active(mark(x0), x1), y1)
MARK(x(x1, x2)) → MARK(x1)
MARK(U12(and(x0, x1), y1)) → U12ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(U11(x0, x1, x2), y1)) → U21ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(and(tt, y1)) → ANDACTIVE(tt, y1)
MARK(U13(x1)) → MARK(x1)
MARK(U41(U13(x0), y1)) → U41ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(and(x0, x1), y1, y2)) → U51ACTIVE(andActive(mark(x0), x1), y1, y2)
XACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U51(U12(x0, x1), y1, y2)) → U51ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U71(x0, x1, x2), y1, y2)) → U71ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U41(x(x0, x1), y1)) → U41ACTIVE(xActive(mark(x0), mark(x1)), y1)
U71ACTIVE(tt, M, N) → MARK(M)
ISNATACTIVE(x(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatKind(V2))
MARK(U12(U21(x0, x1), y1)) → U12ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U12(x1, x2)) → MARK(x1)
MARK(plus(x1, x2)) → MARK(x2)
MARK(U32(U61(x0), y1)) → U32ACTIVE(U61Active(mark(x0)), y1)
MARK(U12(U11(x0, x1, x2), y1)) → U12ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U51(isNat(x0), y1, y2)) → U51ACTIVE(isNatActive(x0), y1, y2)
XACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U41(isNat(x0), y1)) → U41ACTIVE(isNatActive(x0), y1)
MARK(U71(U32(x0, x1), y1, y2)) → U71ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(x(x1, x2)) → XACTIVE(mark(x1), mark(x2))
MARK(U11(tt, y1, y2)) → U11ACTIVE(tt, y1, y2)
MARK(U21(U21(x0, x1), y1)) → U21ACTIVE(U21Active(mark(x0), x1), y1)
MARK(U41(plus(x0, x1), y1)) → U41ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U31(U21(x0, x1), y1, y2)) → U31ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U71(U13(x0), y1, y2)) → U71ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U21(U41(x0, x1), y1)) → U21ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U51(U11(x0, x1, x2), y1, y2)) → U51ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U41(U31(x0, x1, x2), y1)) → U41ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U51(tt, y1, y2)) → U51ACTIVE(tt, y1, y2)
MARK(U32(U12(x0, x1), y1)) → U32ACTIVE(U12Active(mark(x0), x1), y1)
U12ACTIVE(tt, V2) → ISNATACTIVE(V2)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
MARK(U31(U22(x0), y1, y2)) → U31ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → U31ACTIVE(isNatActive(x0), y1, y2)
MARK(and(isNat(x0), y1)) → ANDACTIVE(isNatActive(x0), y1)
MARK(U32(U71(x0, x1, x2), y1)) → U32ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U71(U61(x0), y1, y2)) → U71ACTIVE(U61Active(mark(x0)), y1, y2)
MARK(U71(tt, y1, y2)) → U71ACTIVE(tt, y1, y2)
PLUSACTIVE(N, 0) → ISNATACTIVE(N)
ISNATKINDACTIVE(plus(x(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U33(x1)) → MARK(x1)
MARK(U11(x(x0, x1), y1, y2)) → U11ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
ISNATKINDACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
ISNATKINDACTIVE(x(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U31(x0, x1, x2), y1)) → U21ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(U71(x(x0, x1), y1, y2)) → U71ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
MARK(U41(x1, x2)) → MARK(x1)
MARK(U11(isNat(x0), y1, y2)) → U11ACTIVE(isNatActive(x0), y1, y2)
MARK(U12(U33(x0), y1)) → U12ACTIVE(U33Active(mark(x0)), y1)
ISNATACTIVE(plus(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
PLUSACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U71(U41(x0, x1), y1, y2)) → U71ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(isNatKind(x0), y1)) → U32ACTIVE(isNatKindActive(x0), y1)
MARK(U31(tt, y1, y2)) → U31ACTIVE(tt, y1, y2)
MARK(U21(U22(x0), y1)) → U21ACTIVE(U22Active(mark(x0)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → U32ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(and(U22(x0), y1)) → ANDACTIVE(U22Active(mark(x0)), y1)
MARK(U31(isNatKind(x0), y1, y2)) → U31ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U31(x1, x2, x3)) → MARK(x1)
MARK(U32(U31(x0, x1, x2), y1)) → U32ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(and(U51(x0, x1, x2), y1)) → ANDACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U51(U61(x0), y1, y2)) → U51ACTIVE(U61Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U32(x1, x2)) → MARK(x1)
MARK(U41(U33(x0), y1)) → U41ACTIVE(U33Active(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → U12ACTIVE(U32Active(mark(x0), x1), y1)
XACTIVE(N, 0) → ISNATACTIVE(N)
MARK(U31(U71(x0, x1, x2), y1, y2)) → U31ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → U11ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → U41ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → U11ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(x(x1, x2)) → MARK(x2)
MARK(U11(x1, x2, x3)) → MARK(x1)
MARK(U71(and(x0, x1), y1, y2)) → U71ACTIVE(andActive(mark(x0), x1), y1, y2)
U71ACTIVE(tt, M, N) → PLUSACTIVE(xActive(mark(N), mark(M)), mark(N))
MARK(and(U21(x0, x1), y1)) → ANDACTIVE(U21Active(mark(x0), x1), y1)
U51ACTIVE(tt, M, N) → MARK(N)
MARK(and(x(x0, x1), y1)) → ANDACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(and(U32(x0, x1), y1)) → ANDACTIVE(U32Active(mark(x0), x1), y1)
MARK(U41(tt, y1)) → U41ACTIVE(tt, y1)
MARK(U32(U11(x0, x1, x2), y1)) → U32ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → U12ACTIVE(U12Active(mark(x0), x1), y1)
MARK(U32(tt, y1)) → U32ACTIVE(tt, y1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U32(x(x0, x1), y1)) → U32ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U51(U22(x0), y1, y2)) → U51ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → U71ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U32(U13(x0), y1)) → U32ACTIVE(U13Active(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → U51ACTIVE(xActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, s(M)) → U51ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U11ACTIVE(tt, s(x0), y1) → U12ACTIVE(U21Active(isNatKindActive(x0), x0), y1)
MARK(s(x1)) → MARK(x1)
MARK(U51(plus(x0, x1), y1, y2)) → U51ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(U12(x0, x1), y1, y2)) → U31ACTIVE(U12Active(mark(x0), x1), y1, y2)
U11ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(and(U31(x0, x1, x2), y1)) → ANDACTIVE(U31Active(mark(x0), x1, x2), y1)
ISNATACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U32(U41(x0, x1), y1)) → U32ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U12(U31(x0, x1, x2), y1)) → U12ACTIVE(U31Active(mark(x0), x1, x2), y1)
MARK(and(U61(x0), y1)) → ANDACTIVE(U61Active(mark(x0)), y1)
MARK(U11(U32(x0, x1), y1, y2)) → U11ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U32(U22(x0), y1)) → U32ACTIVE(U22Active(mark(x0)), y1)
MARK(U12(U51(x0, x1, x2), y1)) → U12ACTIVE(U51Active(mark(x0), x1, x2), y1)
MARK(U71(U33(x0), y1, y2)) → U71ACTIVE(U33Active(mark(x0)), y1, y2)
MARK(and(and(x0, x1), y1)) → ANDACTIVE(andActive(mark(x0), x1), y1)
MARK(U12(U61(x0), y1)) → U12ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(x(s(x0), y1)) → ANDACTIVE(isNatKindActive(x0), isNatKind(y1))
MARK(U51(U33(x0), y1, y2)) → U51ACTIVE(U33Active(mark(x0)), y1, y2)
ISNATACTIVE(x(V1, V2)) → U31ACTIVE(andActive(isNatKindActive(V1), isNatKind(V2)), V1, V2)
MARK(U71(U51(x0, x1, x2), y1, y2)) → U71ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U12(isNat(x0), y1)) → U12ACTIVE(isNatActive(x0), y1)
MARK(U11(plus(x0, x1), y1, y2)) → U11ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
PLUSACTIVE(N, 0) → ANDACTIVE(isNatActive(N), isNatKind(N))
MARK(U21(U61(x0), y1)) → U21ACTIVE(U61Active(mark(x0)), y1)
MARK(U11(U21(x0, x1), y1, y2)) → U11ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → ANDACTIVE(U12Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U61(x1)) → MARK(x1)
MARK(U71(U11(x0, x1, x2), y1, y2)) → U71ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → U21ACTIVE(U71Active(mark(x0), x1, x2), y1)
MARK(U31(U33(x0), y1, y2)) → U31ACTIVE(U33Active(mark(x0)), y1, y2)
XACTIVE(N, s(M)) → ANDACTIVE(isNatActive(M), isNatKind(M))
MARK(U31(U13(x0), y1, y2)) → U31ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(and(U33(x0), y1)) → ANDACTIVE(U33Active(mark(x0)), y1)
XACTIVE(N, s(M)) → U71ACTIVE(andActive(andActive(isNatActive(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U32(x0, x1), y1, y2)) → U51ACTIVE(U32Active(mark(x0), x1), y1, y2)
ISNATKINDACTIVE(x(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U31(U11(x0, x1, x2), y1, y2)) → U31ACTIVE(U11Active(mark(x0), x1, x2), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → U31ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → U32ACTIVE(U32Active(mark(x0), x1), y1)
U41ACTIVE(tt, N) → MARK(N)
ISNATKINDACTIVE(plus(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(and(plus(x0, x1), y1)) → ANDACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(plus(x0, x1), y1)) → U32ACTIVE(plusActive(mark(x0), mark(x1)), y1)
U32ACTIVE(tt, V2) → ISNATACTIVE(V2)
MARK(U21(plus(x0, x1), y1)) → U21ACTIVE(plusActive(mark(x0), mark(x1)), y1)
MARK(U32(U21(x0, x1), y1)) → U32ACTIVE(U21Active(mark(x0), x1), y1)
U11ACTIVE(tt, x(x0, x1), y1) → U12ACTIVE(U31Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(tt, y1)) → U21ACTIVE(tt, y1)
MARK(U21(U33(x0), y1)) → U21ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(x1, x2, x3)) → MARK(x1)
PLUSACTIVE(N, s(M)) → ISNATACTIVE(M)
MARK(U41(and(x0, x1), y1)) → U41ACTIVE(andActive(mark(x0), x1), y1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(U31(U61(x0), y1, y2)) → U31ACTIVE(U61Active(mark(x0)), y1, y2)
PLUSACTIVE(N, 0) → U41ACTIVE(andActive(isNatActive(N), isNatKind(N)), N)
U51ACTIVE(tt, M, N) → PLUSACTIVE(mark(N), mark(M))
MARK(U41(U41(x0, x1), y1)) → U41ACTIVE(U41Active(mark(x0), x1), y1)
MARK(U32(and(x0, x1), y1)) → U32ACTIVE(andActive(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(0, y1)) → ANDACTIVE(tt, isNatKind(y1))
MARK(U11(U12(x0, x1), y1, y2)) → U11ACTIVE(U12Active(mark(x0), x1), y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → U71ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(U11(U13(x0), y1, y2)) → U11ACTIVE(U13Active(mark(x0)), y1, y2)
U71ACTIVE(tt, M, N) → XACTIVE(mark(N), mark(M))
MARK(and(isNatKind(x0), y1)) → ANDACTIVE(isNatKindActive(x0), y1)
MARK(U41(U22(x0), y1)) → U41ACTIVE(U22Active(mark(x0)), y1)
U71ACTIVE(tt, M, N) → MARK(N)
MARK(U71(plus(x0, x1), y1, y2)) → U71ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → U31ACTIVE(plusActive(mark(x0), mark(x1)), y1, y2)
MARK(U11(U22(x0), y1, y2)) → U11ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → U51ACTIVE(U21Active(mark(x0), x1), y1, y2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U12(x(x0, x1), y1)) → U12ACTIVE(xActive(mark(x0), mark(x1)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → U41ACTIVE(U11Active(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → U31ACTIVE(U32Active(mark(x0), x1), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → U11ACTIVE(U71Active(mark(x0), x1, x2), y1, y2)
ISNATKINDACTIVE(x(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U21(isNat(x0), y1)) → U21ACTIVE(isNatActive(x0), y1)
MARK(U31(and(x0, x1), y1, y2)) → U31ACTIVE(andActive(mark(x0), x1), y1, y2)
MARK(plus(x1, x2)) → PLUSACTIVE(mark(x1), mark(x2))
MARK(U51(U41(x0, x1), y1, y2)) → U51ACTIVE(U41Active(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → U32ACTIVE(U33Active(mark(x0)), y1)
MARK(U71(isNat(x0), y1, y2)) → U71ACTIVE(isNatActive(x0), y1, y2)
MARK(U11(U51(x0, x1, x2), y1, y2)) → U11ACTIVE(U51Active(mark(x0), x1, x2), y1, y2)
U11ACTIVE(tt, plus(x0, x1), y1) → U12ACTIVE(U11Active(andActive(isNatKindActive(x0), isNatKind(x1)), x0, x1), y1)
MARK(U41(U32(x0, x1), y1)) → U41ACTIVE(U32Active(mark(x0), x1), y1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U41(U21(x0, x1), y1)) → U41ACTIVE(U21Active(mark(x0), x1), y1)
ISNATKINDACTIVE(plus(plus(x0, x1), y1)) → ANDACTIVE(andActive(isNatKindActive(x0), isNatKind(x1)), isNatKind(y1))
MARK(U71(U22(x0), y1, y2)) → U71ACTIVE(U22Active(mark(x0)), y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → U31ACTIVE(U31Active(mark(x0), x1, x2), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → U51ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U12(U13(x0), y1)) → U12ACTIVE(U13Active(mark(x0)), y1)
MARK(U71(U12(x0, x1), y1, y2)) → U71ACTIVE(U12Active(mark(x0), x1), y1, y2)
U31ACTIVE(tt, V1, V2) → U32ACTIVE(isNatActive(V1), V2)
MARK(U51(U13(x0), y1, y2)) → U51ACTIVE(U13Active(mark(x0)), y1, y2)
MARK(U11(isNatKind(x0), y1, y2)) → U11ACTIVE(isNatKindActive(x0), y1, y2)
MARK(U21(U13(x0), y1)) → U21ACTIVE(U13Active(mark(x0)), y1)
MARK(U41(U61(x0), y1)) → U41ACTIVE(U61Active(mark(x0)), y1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)

The TRS R consists of the following rules:

mark(U11(x1,