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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
↳15 QTRSRRRProof (⇔)
↳16 QTRS
↳17 QTRSRRRProof (⇔)
↳18 QTRS
  ↳19 RisEmptyProof (⇔)
    ↳20 TRUE
  ↳21 RisEmptyProof (⇔)
    ↳22 TRUE

(0) Obligation:

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

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U21(tt, M, N) → a__U22(tt, M, N)
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
a__U11(x1, x2, x3)  =  a__U11(x1, x2, x3)
tt  =  tt
a__U12(x1, x2, x3)  =  a__U12(x1, x2, x3)
s(x1)  =  s(x1)
a__plus(x1, x2)  =  a__plus(x1, x2)
mark(x1)  =  x1
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
a__U22(x1, x2, x3)  =  a__U22(x1, x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
0  =  0
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)

Recursive path order with status [RPO].
Quasi-Precedence:
[aU213, aU223, ax2, U213, U223, x2] > [aU113, aU123, aplus2, U113, U123, plus2] > s1 > tt

Status:
plus2: [1,2]
U113: [3,2,1]
aU213: multiset
x2: multiset
aU123: [3,2,1]
0: multiset
ax2: multiset
aplus2: [1,2]
aU223: multiset
tt: multiset
aU113: [3,2,1]
U223: multiset
s1: multiset
U123: [3,2,1]
U213: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U22(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
a__x(N, 0) → 0
a__x(N, s(M)) → a__U21(tt, M, N)


(2) Obligation:

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

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U21(tt, M, N) → a__U22(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(U21(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U11(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U21(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__plus(x1, x2)) = x1 + x2   
POL(a__x(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(x(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U21(tt, M, N) → a__U22(tt, M, N)


(4) Obligation:

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

a__U11(tt, M, N) → a__U12(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(U21(x1, x2, x3)) = x1 + x2 + x3   
POL(U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U11(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U21(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__plus(x1, x2)) = x1 + x2   
POL(a__x(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(x(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U11(tt, M, N) → a__U12(tt, M, N)


(6) Obligation:

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

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(U21(x1, x2, x3)) = 1 + x1 + x2 + 2·x3   
POL(U22(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(a__U11(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(a__U12(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(a__U21(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(a__U22(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(a__plus(x1, x2)) = 2 + x1 + x2   
POL(a__x(x1, x2)) = x1 + 2·x2   
POL(mark(x1)) = 2·x1   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = 2 + x1   
POL(tt) = 2   
POL(x(x1, x2)) = x1 + 2·x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__plus(X1, X2) → plus(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)


(8) Obligation:

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

mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(U21(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U11(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U21(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__plus(x1, x2)) = x1 + x2   
POL(a__x(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(x(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
a__U11(X1, X2, X3) → U11(X1, X2, X3)


(10) Obligation:

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

mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U12(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + x3   
POL(U22(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(a__U12(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(a__U22(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(a__x(x1, x2)) = x1 + 2·x2   
POL(mark(x1)) = 2·x1   
POL(x(x1, x2)) = x1 + 2·x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)


(12) Obligation:

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

mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U12(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U22(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U22(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__x(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(x(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)


(14) Obligation:

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

mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
a__x(X1, X2) → x(X1, X2)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__x(x1, x2)) = 2 + x1 + 2·x2   
POL(mark(x1)) = 2·x1   
POL(x(x1, x2)) = 1 + x1 + 2·x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__x(X1, X2) → x(X1, X2)


(16) Obligation:

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

mark(x(X1, X2)) → a__x(mark(X1), mark(X2))

Q is empty.

(17) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__x(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(x(x1, x2)) = 1 + x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(x(X1, X2)) → a__x(mark(X1), mark(X2))


(18) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(19) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(20) TRUE

(21) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(22) TRUE