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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
    ↳5 QDP
        ↳6 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 PisEmptyProof (⇔)
        ↳9 TRUE
    ↳10 QDP
        ↳11 QDPOrderProof (⇔)
        ↳12 QDP
        ↳13 PisEmptyProof (⇔)
        ↳14 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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(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)
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U211(tt, M, N) → ACTIVATE(M)
U211(tt, M, N) → ACTIVATE(N)
U221(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U221(tt, M, N) → X(activate(N), activate(M))
U221(tt, M, N) → ACTIVATE(N)
U221(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U111(tt, M, N)
X(N, s(M)) → U211(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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(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 2 SCCs with 9 less nodes.

(4) Complex Obligation (AND)

(5) 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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

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

(6) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
U121(x0, x1, x2, x3)  =  U121(x0)
PLUS(x0, x1, x2)  =  PLUS(x0, x1)
U111(x0, x1, x2, x3)  =  U111(x0, x1, x3)

Tags:
U121 has argument tags [3,7,7,8] and root tag 1
PLUS has argument tags [0,0,8] and root tag 3
U111 has argument tags [2,15,2,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U121(x1, x2, x3)  =  U121(x2, x3)
tt  =  tt
PLUS(x1, x2)  =  PLUS(x1, x2)
activate(x1)  =  x1
s(x1)  =  s(x1)
U111(x1, x2, x3)  =  U111(x1, x2, x3)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U12^12, tt, PLUS2, U11^13]

Status:
U12^12: [2,1]
tt: []
PLUS2: [1,2]
s1: [1]
U11^13: [3,2,1]


The following usable rules [FROCOS05] were oriented:

activate(X) → X

(7) 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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(tt, activate(M), activate(N))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U221(x0, x1, x2, x3)  =  U221(x0, x1)
X(x0, x1, x2)  =  X(x0)
U211(x0, x1, x2, x3)  =  U211(x0, x1)

Tags:
U221 has argument tags [11,3,0,13] and root tag 2
X has argument tags [4,0,12] and root tag 3
U211 has argument tags [11,11,0,12] and root tag 0

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U221(x1, x2, x3)  =  U221(x2, x3)
tt  =  tt
X(x1, x2)  =  X(x1, x2)
activate(x1)  =  x1
s(x1)  =  s(x1)
U211(x1, x2, x3)  =  U211(x2, x3)

Lexicographic path order with status [LPO].
Quasi-Precedence:
s1 > tt > [U22^12, X2, U21^12]

Status:
U22^12: [2,1]
tt: []
X2: [1,2]
s1: [1]
U21^12: [2,1]


The following usable rules [FROCOS05] were oriented:

activate(X) → X

(12) 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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE