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

(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) 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, M, N) → U12¹(tt, activate(M), activate(N))
U11¹(tt, M, N) → ACTIVATE(M)
U11¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, M, N) → PLUS(activate(N), activate(M))
U12¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U11¹(tt, M, N)

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.
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 4 less nodes.

(4) Obligation:

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

U12¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U11¹(tt, M, N)
U11¹(tt, M, N) → U12¹(tt, activate(M), activate(N))

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

(5) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation [POLO]:

POL(activate(x₁)) = x₁   
POL(s(x₁)) = 1 + x₁   
POL(tt) = 1   

From the DPs we obtained the following set of size-change graphs:

• PLUS(N, s(M)) → U11¹(tt, M, N) (allowed arguments on rhs = {2, 3})
  The graph contains the following edges 2 > 2, 1 >= 3

• U11¹(tt, M, N) → U12¹(tt, activate(M), activate(N)) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 >= 1, 2 >= 2

• U12¹(tt, M, N) → PLUS(activate(N), activate(M)) (allowed arguments on rhs = {2})
  The graph contains the following edges 2 >= 2

We oriented the following set of usable rules [AAECC05,FROCOS05].

activate(X) → X

(6) TRUE