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) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
activate(x0)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

(4) 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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
activate(x0)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(6) 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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
activate(x0)

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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 remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x1, x2)
tt  =  tt
PLUS(x1, x2)  =  PLUS(x2)
activate(x1)  =  x1
s(x1)  =  s(x1)
U11¹(x1, x2, x3)  =  U11¹(x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
0  =  0

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

[U11₃, U12₃, plus₂] > s₁ > [tt, U11^1₁] > U12^1₂ > PLUS₁

Status:

U12^1₂: [2,1]
tt: multiset
PLUS₁: multiset
s₁: [1]
U11^1₁: [1]
U11₃: [2,3,1]
U12₃: [2,3,1]
plus₂: [2,1]
0: multiset

The following usable rules [FROCOS05] were oriented:

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

(8) Obligation:

Q DP problem:
P is empty.
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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
activate(x0)

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

(9) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(10) TRUE