Termination w.r.t. Q proof of /tmp/tmpIy0Hah/MYNAT_complete

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 DependencyGraphProof (⇔)

↳12 TRUE

(0) Obligation:

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(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(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)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

(4) Obligation:

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

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

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(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(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)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[n_{_x₂}, X₂, U31^1₂, U71^1₃, x₂, U71₃] > [n_{_plus₂}, U11^1₂, plus₂, U51₃] > U12^1₂ > [tt, ACTIVATE₁, 0, AND₁, n_{_s₁}, U21^1₁, s₁, n_₀, U22] > ISNATKIND₁ > [isNatKind₁, n_{_isNatKind₁}]
[n_{_x₂}, X₂, U31^1₂, U71^1₃, x₂, U71₃] > [n_{_plus₂}, U11^1₂, plus₂, U51₃] > [isNat, U31, U11, U33, U32, U12, U13] > U21 > [tt, ACTIVATE₁, 0, AND₁, n_{_s₁}, U21^1₁, s₁, n_₀, U22] > ISNATKIND₁ > [isNatKind₁, n_{_isNatKind₁}]
[n_{_x₂}, X₂, U31^1₂, U71^1₃, x₂, U71₃] > [n_{_plus₂}, U11^1₂, plus₂, U51₃] > [PLUS₂, U51^1₂] > U41^1₂ > [tt, ACTIVATE₁, 0, AND₁, n_{_s₁}, U21^1₁, s₁, n_₀, U22] > ISNATKIND₁ > [isNatKind₁, n_{_isNatKind₁}]
[n_{_x₂}, X₂, U31^1₂, U71^1₃, x₂, U71₃] > U32^1₂ > [tt, ACTIVATE₁, 0, AND₁, n_{_s₁}, U21^1₁, s₁, n_₀, U22] > ISNATKIND₁ > [isNatKind₁, n_{_isNatKind₁}]

Status:

n_{_plus₂}: [2,1]
U51^1₂: [2,1]
U12^1₂: [1,2]
U22: []
U31: []
U11: []
x₂: [2,1]
U33: []
n_{_s₁}: [1]
isNat: []
U71^1₃: [2,3,1]
PLUS₂: [1,2]
tt: []
U11^1₂: [2,1]
U31^1₂: [2,1]
s₁: [1]
U51₃: [2,3,1]
ACTIVATE₁: [1]
U13: []
X₂: [2,1]
plus₂: [2,1]
U21: []
AND₁: [1]
U32: []
n_{_isNatKind₁}: [1]
ISNATKIND₁: [1]
U32^1₂: [1,2]
U12: []
isNatKind₁: [1]
0: []
U41^1₂: [1,2]
n_₀: []
n_{_x₂}: [2,1]
U21^1₁: [1]
U71₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1) → ACTIVATE(V1)
U51¹(tt, M, N) → PLUS(activate(N), activate(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(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(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)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

(8) Obligation:

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

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

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(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(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)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = ACTIVATE(x1)
n__and(x1, x2) = n__and(x2)
AND(x1, x2) = AND(x2)
tt = tt

Lexicographic path order with status [LPO].
Quasi-Precedence:

[ACTIVATE₁, n_{_and₁}, AND₁, tt]

Status:

n_{_and₁}: [1]
tt: []
AND₁: [1]
ACTIVATE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

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

AND(tt, X) → ACTIVATE(X)

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(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(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)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) TRUE