(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(N, s(M)) → U111(tt, M, N)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2, x3)  =  x2
tt  =  tt
PLUS(x1, x2)  =  x2
activate(x1)  =  x1
s(x1)  =  s(x1)
U111(x1, x2, x3)  =  x2
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U213, U223, x2] > [tt, U113, U123, plus2] > s1 > 0

Status:
tt: []
s1: [1]
U113: [2,3,1]
U123: [2,3,1]
plus2: [2,1]
U213: [3,2,1]
U223: [3,2,1]
x2: [1,2]
0: []


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)))
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

(9) Obligation:

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

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

(10) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(11) TRUE

(12) 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.

(13) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1, x2, x3)  =  U221(x1, x2)
tt  =  tt
X(x1, x2)  =  x2
activate(x1)  =  x1
s(x1)  =  s(x1)
U211(x1, x2, x3)  =  U211(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U213, U223, x2] > [U113, U123, plus2] > s1 > [U22^12, tt, U21^12]

Status:
U22^12: [2,1]
tt: []
s1: [1]
U21^12: [2,1]
U113: [3,2,1]
U123: [3,2,1]
plus2: [1,2]
U213: [3,2,1]
U223: [3,2,1]
x2: [1,2]
0: []


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)))
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

(14) Obligation:

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

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.

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE