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

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 UsableRulesProof (⇔)
    ↳9 QDP
    ↳10 QReductionProof (⇔)
    ↳11 QDP
    ↳12 UsableRulesReductionPairsProof (⇔)
    ↳13 QDP
    ↳14 DependencyGraphProof (⇔)
    ↳15 TRUE
  ↳16 QDP
    ↳17 UsableRulesProof (⇔)
    ↳18 QDP
    ↳19 QReductionProof (⇔)
    ↳20 QDP
    ↳21 UsableRulesReductionPairsProof (⇔)
    ↳22 QDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 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) 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)))
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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(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)
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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(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 2 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) 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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)

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

(8) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(9) 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:

activate(X) → X

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)

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

(10) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))

(11) 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:

activate(X) → X

The set Q consists of the following terms:

activate(x0)

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

(12) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

PLUS(N, s(M)) → U111(tt, M, N)
U111(tt, M, N) → U121(tt, activate(M), activate(N))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(PLUS(x1, x2)) = x1 + x2   
POL(U111(x1, x2, x3)) = 1 + 2·x1 + x2 + x3   
POL(U121(x1, x2, x3)) = x1 + x2 + x3   
POL(activate(x1)) = x1   
POL(s(x1)) = 2 + 2·x1   
POL(tt) = 0   

(13) Obligation:

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

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

The TRS R consists of the following rules:

activate(X) → X

The set Q consists of the following terms:

activate(x0)

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

(14) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(15) TRUE

(16) 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

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)

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

(17) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(18) 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:

activate(X) → X

The set Q consists of the following terms:

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)

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

(19) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))

(20) 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:

activate(X) → X

The set Q consists of the following terms:

activate(x0)

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

(21) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

X(N, s(M)) → U211(tt, M, N)
U211(tt, M, N) → U221(tt, activate(M), activate(N))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(U211(x1, x2, x3)) = 1 + 2·x1 + x2 + x3   
POL(U221(x1, x2, x3)) = x1 + x2 + x3   
POL(X(x1, x2)) = x1 + x2   
POL(activate(x1)) = x1   
POL(s(x1)) = 2 + 2·x1   
POL(tt) = 0   

(22) Obligation:

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

U221(tt, M, N) → X(activate(N), activate(M))

The TRS R consists of the following rules:

activate(X) → X

The set Q consists of the following terms:

activate(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE