Termination w.r.t. Q proof of /tmp/tmplJ4XoG/MYNAT_complete

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

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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:

U11¹(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U51¹(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
U51¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
X(N, 0) → ISNAT(N)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 0¹
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U31¹(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__s(X)) → S(activate(X))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U71¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → U51¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U12¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U51¹(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNAT(n__x(V1, V2)) → U31¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U51¹(tt, M, N) → S(plus(activate(N), activate(M)))
U31¹(tt, V1, V2) → ISNAT(activate(V1))
U12¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
U61¹(tt) → 0¹
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
U71¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U32¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V1, V2) → U32¹(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U12¹(tt, V2) → U13¹(isNat(activate(V2)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U21¹(tt, V1) → ACTIVATE(V1)
X(N, 0) → U61¹(and(isNat(N), n__isNatKind(N)))
U31¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U71¹(tt, M, N) → X(activate(N), activate(M))
U32¹(tt, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
U32¹(tt, V2) → U33¹(isNat(activate(V2)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

U11¹(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U51¹(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
U51¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
X(N, 0) → ISNAT(N)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 0¹
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U31¹(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__s(X)) → S(activate(X))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U71¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → U51¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U12¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U51¹(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNAT(n__x(V1, V2)) → U31¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U51¹(tt, M, N) → S(plus(activate(N), activate(M)))
U31¹(tt, V1, V2) → ISNAT(activate(V1))
U12¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
U61¹(tt) → 0¹
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
U71¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U32¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V1, V2) → U32¹(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U12¹(tt, V2) → U13¹(isNat(activate(V2)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U21¹(tt, V1) → ACTIVATE(V1)
X(N, 0) → U61¹(and(isNat(N), n__isNatKind(N)))
U31¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U71¹(tt, M, N) → X(activate(N), activate(M))
U32¹(tt, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
U32¹(tt, V2) → U33¹(isNat(activate(V2)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

U11¹(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
X(N, 0) → ISNAT(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U31¹(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U71¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → U51¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U12¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U51¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNAT(n__x(V1, V2)) → U31¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U31¹(tt, V1, V2) → ISNAT(activate(V1))
U12¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
U71¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U31¹(tt, V1, V2) → U32¹(isNat(activate(V1)), activate(V2))
U32¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U21¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U71¹(tt, M, N) → X(activate(N), activate(M))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U32¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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.

U11¹(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U51¹(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
X(N, 0) → ISNAT(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U31¹(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U71¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → U51¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U21¹(tt, V1) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U51¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNAT(n__x(V1, V2)) → U31¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
X(N, s(M)) → U71¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U31¹(tt, V1, V2) → ISNAT(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
U71¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U32¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U21¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U71¹(tt, M, N) → X(activate(N), activate(M))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U32¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))

The remaining pairs can at least be oriented weakly.

U51¹(tt, M, N) → PLUS(activate(N), activate(M))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U12¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V1, V2) → U32¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)

Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3) = U11¹(x2, x3)
tt = tt
ACTIVATE(x1) = x1
ISNAT(x1) = x1
n__x(x1, x2) = n__x(x1, x2)
AND(x1, x2) = x2
isNatKind(x1) = x1
activate(x1) = x1
n__isNatKind(x1) = x1
U51¹(x1, x2, x3) = U51¹(x2, x3)
PLUS(x1, x2) = PLUS(x1, x2)
0 = 0
isNat(x1) = x1
X(x1, x2) = X(x1, x2)
n__s(x1) = n__s(x1)
U12¹(x1, x2) = x2
n__plus(x1, x2) = n__plus(x1, x2)
and(x1, x2) = and(x1, x2)
ISNATKIND(x1) = x1
s(x1) = s(x1)
U31¹(x1, x2, x3) = U31¹(x2, x3)
U41¹(x1, x2) = U41¹(x2)
U71¹(x1, x2, x3) = U71¹(x1, x2, x3)
n__and(x1, x2) = n__and(x1, x2)
n__isNat(x1) = x1
U21¹(x1, x2) = U21¹(x2)
x(x1, x2) = x(x1, x2)
U32¹(x1, x2) = U32¹(x1, x2)
U11(x1, x2, x3) = U11(x1)
U12(x1, x2) = U12
U13(x1) = U13
U22(x1) = U22
U21(x1, x2) = x1
n__0 = n__0
U31(x1, x2, x3) = U31(x1, x2, x3)
U32(x1, x2) = x2
U33(x1) = x1
U51(x1, x2, x3) = U51(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U61(x1) = U61(x1)
U71(x1, x2, x3) = U71(x1, x2, x3)
U41(x1, x2) = U41(x1, x2)

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

[tt, 0, U12, U13, U22, n_₀] > [n_{_x₂}, X₂, U71^1₃, x₂, U71₃] > [U11^1₂, n_{_plus₂}, U51₃, plus₂] > [U51^1₂, PLUS₂] > [and₂, n_{_and₂}] > [U11₁, U31₃]
[tt, 0, U12, U13, U22, n_₀] > [n_{_x₂}, X₂, U71^1₃, x₂, U71₃] > [U11^1₂, n_{_plus₂}, U51₃, plus₂] > [U51^1₂, PLUS₂] > U41^1₁ > [U11₁, U31₃]
[tt, 0, U12, U13, U22, n_₀] > [n_{_x₂}, X₂, U71^1₃, x₂, U71₃] > [U11^1₂, n_{_plus₂}, U51₃, plus₂] > [n_{_s₁}, s₁, U21^1₁] > [U11₁, U31₃]
[tt, 0, U12, U13, U22, n_₀] > [n_{_x₂}, X₂, U71^1₃, x₂, U71₃] > [U11^1₂, n_{_plus₂}, U51₃, plus₂] > U41₂ > [U11₁, U31₃]
[tt, 0, U12, U13, U22, n_₀] > [n_{_x₂}, X₂, U71^1₃, x₂, U71₃] > [U31^1₂, U32^1₂] > [U11₁, U31₃]
[tt, 0, U12, U13, U22, n_₀] > [n_{_x₂}, X₂, U71^1₃, x₂, U71₃] > U61₁ > [U11₁, U31₃]

Status:

n_{_plus₂}: [1,2]
U51^1₂: multiset
U41^1₁: multiset
U22: []
x₂: [2,1]
and₂: multiset
n_{_s₁}: multiset
U71^1₃: [2,3,1]
PLUS₂: multiset
U31₃: multiset
tt: multiset
U11^1₂: [1,2]
U31^1₂: multiset
U11₁: multiset
s₁: multiset
U51₃: [3,2,1]
U13: []
X₂: [2,1]
plus₂: [1,2]
U61₁: multiset
U12: []
U32^1₂: multiset
0: multiset
n_{_and₂}: multiset
U41₂: multiset
n_₀: multiset
n_{_x₂}: [2,1]
U21^1₁: multiset
U71₃: [2,3,1]

The following usable rules [17] were oriented:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
activate(n__0) → 0
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
x(X1, X2) → n__x(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
activate(n__isNatKind(X)) → isNatKind(X)
U41(tt, N) → activate(N)
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
s(X) → n__s(X)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
0 → n__0
activate(X) → X
activate(n__isNat(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

U12¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
U12¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U31¹(tt, V1, V2) → U32¹(isNat(activate(V1)), activate(V2))
AND(tt, X) → ACTIVATE(X)
U51¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 10 less nodes.