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 RisEmptyProof (⇔)
↳10 TRUE

(0) Obligation:

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

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(activate(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

plus(N, 0) → N


(2) Obligation:

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

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(U12(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(activate(x1)) = x1   
POL(plus(x1, x2)) = 2·x1 + 2·x2   
POL(s(x1)) = 1 + x1   
POL(tt) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

plus(N, s(M)) → U11(tt, M, N)


(4) Obligation:

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

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(U12(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(activate(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 2·x1   
POL(tt) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))


(6) Obligation:

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

activate(X) → X

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

activate(X) → X


(8) Obligation:

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

(9) RisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE