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

(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)))
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))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(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)
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(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)
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)
0  =  0

Recursive path order with status [RPO].
Quasi-Precedence:
[U213, plus2, nplus2] > U112 > [tt, and2, isNat1, n0, nisNat1, 0]
[U213, plus2, nplus2] > [s1, ns1] > [tt, and2, isNat1, n0, nisNat1, 0]

Status:
nplus2: [1,2]
plus2: [1,2]
nisNat1: multiset
U112: multiset
and2: [1,2]
ns1: [1]
0: multiset
tt: multiset
n0: multiset
s1: [1]
isNat1: multiset
U213: [3,2,1]

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)))
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))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)


(2) 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)
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(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(activate(x1)) = 1 + 2·x1   
POL(isNat(x1)) = 1 + 2·x1   
POL(n__0) = 0   
POL(n__isNat(x1)) = 1 + x1   
POL(n__plus(x1, x2)) = 2 + 2·x1 + x2   
POL(n__s(x1)) = 1 + x1   
POL(plus(x1, x2)) = 2 + 2·x1 + x2   
POL(s(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

0 → n__0
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(X) → X


(4) Obligation:

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

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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(activate(x1)) = x1   
POL(isNat(x1)) = 1 + x1   
POL(n__0) = 1   
POL(n__isNat(x1)) = x1   
POL(n__plus(x1, x2)) = x1 + x2   
POL(n__s(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0


(6) 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.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(activate(x1)) = 2·x1   
POL(n__plus(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(plus(x1, x2)) = 1 + 2·x1 + 2·x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(8) Obligation:

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

plus(X1, X2) → n__plus(X1, X2)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(n__plus(x1, x2)) = 1 + x1 + x2   
POL(plus(x1, x2)) = 2 + 2·x1 + 2·x2   
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)


(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