(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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:

ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
ACTIVE(U11(tt, M, N)) → U121(tt, M, N)
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(U12(tt, M, N)) → S(plus(N, M))
ACTIVE(U12(tt, M, N)) → PLUS(N, M)
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
ACTIVE(U21(tt, M, N)) → U221(tt, M, N)
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U22(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U22(tt, M, N)) → X(N, M)
ACTIVE(plus(N, 0)) → MARK(N)
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
ACTIVE(plus(N, s(M))) → U111(tt, M, N)
ACTIVE(x(N, 0)) → MARK(0)
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
ACTIVE(x(N, s(M))) → U211(tt, M, N)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U111(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → U121(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U21(X1, X2, X3)) → U211(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
MARK(U22(X1, X2, X3)) → U221(mark(X1), X2, X3)
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(x(X1, X2)) → X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)
U121(mark(X1), X2, X3) → U121(X1, X2, X3)
U121(X1, mark(X2), X3) → U121(X1, X2, X3)
U121(X1, X2, mark(X3)) → U121(X1, X2, X3)
U121(active(X1), X2, X3) → U121(X1, X2, X3)
U121(X1, active(X2), X3) → U121(X1, X2, X3)
U121(X1, X2, active(X3)) → U121(X1, X2, X3)
S(mark(X)) → S(X)
S(active(X)) → S(X)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(X1, X2, mark(X3)) → U211(X1, X2, X3)
U211(active(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)
U211(X1, X2, active(X3)) → U211(X1, X2, X3)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
X(mark(X1), X2) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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 8 SCCs with 18 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(6) Obligation:

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

U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(7) Obligation:

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

U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, X2, mark(X3)) → U211(X1, X2, X3)
U211(active(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)
U211(X1, X2, active(X3)) → U211(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(8) Obligation:

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

PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(9) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(10) Obligation:

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

U121(X1, mark(X2), X3) → U121(X1, X2, X3)
U121(mark(X1), X2, X3) → U121(X1, X2, X3)
U121(X1, X2, mark(X3)) → U121(X1, X2, X3)
U121(active(X1), X2, X3) → U121(X1, X2, X3)
U121(X1, active(X2), X3) → U121(X1, X2, X3)
U121(X1, X2, active(X3)) → U121(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(11) Obligation:

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

U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(12) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
ACTIVE(plus(N, 0)) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

Q is empty.
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.


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

Recursive Path Order [RPO].
Precedence:
[MARK, U11, mark, tt, U12, plus, U21, U22, x, active] > s > 0


The following usable rules [FROCOS05] were oriented:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(14) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
ACTIVE(plus(N, 0)) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(U12(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
ACTIVE(U22(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(plus(N, 0)) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
ACTIVE(plus(N, s(M))) → MARK(U11(tt, M, N))
MARK(U22(X1, X2, X3)) → MARK(X1)
ACTIVE(x(N, s(M))) → MARK(U21(tt, M, N))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  U11(x1, x2, x3)
ACTIVE(x1)  =  x1
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x2, x3)
s(x1)  =  s(x1)
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
active(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
[U213, U223, x2] > [U113, U123, plus2] > s1 > tt
[U213, U223, x2] > 0 > tt


The following usable rules [FROCOS05] were oriented:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(16) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
U21(x1, x2, x3)  =  U21
U22(x1, x2, x3)  =  U22
plus(x1, x2)  =  plus
x(x1, x2)  =  x
active(x1)  =  active
s(x1)  =  s
0  =  0

Recursive Path Order [RPO].
Precedence:
[x, 0] > [mark, active] > U21 > tt > [U11, ACTIVE, U12, U22, plus, s]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(18) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
U21(x1, x2, x3)  =  U21
U22(x1, x2, x3)  =  U22
plus(x1, x2)  =  plus
active(x1)  =  active
s(x1)  =  s
x(x1, x2)  =  x
0  =  0

Recursive Path Order [RPO].
Precedence:
U21 > [U11, ACTIVE, U12, U22] > [mark, active]
plus > tt > s > [U11, ACTIVE, U12, U22] > [mark, active]
plus > tt > [x, 0] > [mark, active]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(20) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U21(tt, M, N)) → MARK(U22(tt, M, N))
MARK(U22(X1, X2, X3)) → ACTIVE(U22(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
U21(x1, x2, x3)  =  U21
U22(x1, x2, x3)  =  U22
active(x1)  =  active
s(x1)  =  s
plus(x1, x2)  =  plus
x(x1, x2)  =  x
0  =  0

Recursive Path Order [RPO].
Precedence:
U21 > [MARK, U11] > U12 > [mark, tt, active, s, x]
U21 > [MARK, U11] > U22 > plus > [mark, tt, active, s, x]
0 > [mark, tt, active, s, x]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(22) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))

The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, M, N)) → MARK(U12(tt, M, N))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
active(x1)  =  active
s(x1)  =  s
plus(x1, x2)  =  plus
U21(x1, x2, x3)  =  U21
U22(x1, x2, x3)  =  U22
x(x1, x2)  =  x
0  =  0

Recursive Path Order [RPO].
Precedence:
[mark, active] > tt > s > U11 > MARK1
[mark, active] > tt > s > U11 > U12
[mark, active] > tt > s > U21
[mark, active] > tt > plus > U11 > MARK1
[mark, active] > tt > plus > U11 > U12
[mark, active] > tt > U22
[mark, active] > tt > x > U21
[mark, active] > tt > x > 0


The following usable rules [FROCOS05] were oriented:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

(24) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(X1, X2, X3)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE