Termination w.r.t. Q proof of /tmp/tmpjNRhhE/MYNAT_nokinds

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

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 QTRSRRRProof (⇔)

↳6 QTRS

  ↳7 QTRSRRRProof (⇔)

    ↳8 QTRS

    ↳9 QTRSRRRProof (⇔)

    ↳10 QTRS

      ↳11 RisEmptyProof (⇔)

        ↳12 TRUE

      ↳13 RisEmptyProof (⇔)

        ↳14 TRUE

      ↳15 RisEmptyProof (⇔)

        ↳16 TRUE

  ↳17 QTRSRRRProof (⇔)

    ↳18 QTRS

(0) Obligation:

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
U11(x1, x2) = U11(x1, x2)
tt = tt
activate(x1) = x1
U21(x1, x2, x3) = U21(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
U31(x1) = x1
0 = 0
U41(x1, x2, x3) = U41(x1, x2, x3)
x(x1, x2) = x(x1, x2)
and(x1, x2) = and(x1, x2)
isNat(x1) = isNat(x1)
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
n__isNat(x1) = n__isNat(x1)
n__s(x1) = n__s(x1)
n__x(x1, x2) = n__x(x1, x2)

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

[tt, 0, n_₀] > [U21₃, plus₂, n_{_plus₂}] > U11₂ > [isNat₁, n_{_isNat₁}]
[tt, 0, n_₀] > [U21₃, plus₂, n_{_plus₂}] > [s₁, n_{_s₁}] > and₂ > [isNat₁, n_{_isNat₁}]
[U41₃, x₂, n_{_x₂}] > [U21₃, plus₂, n_{_plus₂}] > U11₂ > [isNat₁, n_{_isNat₁}]
[U41₃, x₂, n_{_x₂}] > [U21₃, plus₂, n_{_plus₂}] > [s₁, n_{_s₁}] > and₂ > [isNat₁, n_{_isNat₁}]

Status:

n_{_plus₂}: [2,1]
plus₂: [2,1]
n_{_isNat₁}: multiset
U41₃: [3,2,1]
x₂: [1,2]
and₂: multiset
U11₂: multiset
n_{_s₁}: [1]
0: multiset
tt: multiset
n_₀: multiset
n_{_x₂}: [1,2]
s₁: [1]
isNat₁: multiset
U21₃: [2,3,1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)

(2) Obligation:

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

U31(tt) → 0
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U31(x₁)) = 1 + x₁
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = x₁
POL(n__0) = 0
POL(n__isNat(x₁)) = x₁
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__x(x₁, x₂)) = x₁ + x₂
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0
POL(x(x₁, x₂)) = x₁ + x₂

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U31(tt) → 0

(4) Obligation:

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

0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 2
POL(activate(x₁)) = 1 + 2·x₁
POL(isNat(x₁)) = 2 + x₁
POL(n__0) = 2
POL(n__isNat(x₁)) = 2 + x₁
POL(n__plus(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(n__x(x₁, x₂)) = 2 + x₁ + x₂
POL(plus(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(s(x₁)) = 2 + x₁
POL(x(x₁, x₂)) = 2 + x₁ + x₂

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

s(X) → n__s(X)
activate(n__0) → 0
activate(n__isNat(X)) → isNat(X)
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

(6) Obligation:

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

0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
x(X1, X2) → n__x(X1, X2)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
0 = 0
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
n__plus(x1, x2) = n__plus(x1, x2)
isNat(x1) = isNat(x1)
n__isNat(x1) = n__isNat(x1)
x(x1, x2) = x(x1, x2)
n__x(x1, x2) = n__x(x1, x2)
activate(x1) = activate(x1)
n__s(x1) = n__s(x1)
s(x1) = x1

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

[0, n_₀] > n_{_isNat₁}
isNat₁ > n_{_isNat₁}
x₂ > n_{_x₂} > n_{_isNat₁}
activate₁ > plus₂ > n_{_plus₂} > n_{_isNat₁}
n_{_s₁} > n_{_isNat₁}

Status:

n_{_plus₂}: [2,1]
plus₂: multiset
n_{_isNat₁}: multiset
n_₀: multiset
n_{_x₂}: multiset
x₂: [2,1]
activate₁: multiset
n_{_s₁}: multiset
isNat₁: multiset
0: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
x(X1, X2) → n__x(X1, X2)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))

(8) Obligation:

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

0 → n__0

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(n__0) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

0 → n__0

(10) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(11) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(12) TRUE

(13) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(14) TRUE

(15) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(16) TRUE

(17) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__isNat(x₁)) = x₁
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(n__x(x₁, x₂)) = x₁ + x₂
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(x(x₁, x₂)) = 1 + x₁ + x₂

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

0 → n__0
isNat(X) → n__isNat(X)
x(X1, X2) → n__x(X1, X2)
activate(n__s(X)) → s(activate(X))

(18) Obligation:

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

plus(X1, X2) → n__plus(X1, X2)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))

Q is empty.