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 RisEmptyProof (⇔)
    ↳18 TRUE
  ↳19 RisEmptyProof (⇔)
    ↳20 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__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(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(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)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(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__plus(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__plus(N, 0) → mark(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__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, s(M)) → a__U11(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(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)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

a__plus(N, s(M)) → a__U11(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)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
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(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)

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(a__U11(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = x1 + x2 + x3   
POL(a__plus(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 1   
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)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))


(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(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)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1, x2, x3)) = 1 + 2·x1 + x2 + x3   
POL(U12(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U11(x1, x2, x3)) = 2 + 2·x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = 1 + x1 + 2·x2 + 2·x3   
POL(a__plus(x1, x2)) = x1 + x2   
POL(mark(x1)) = 2·x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 1 + 2·x1   
POL(tt) = 1   
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)
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__U11(X1, X2, X3) → U11(X1, X2, X3)


(8) 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(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(0) → 0
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Q is empty.

(9) 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(a__U11(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U12(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__plus(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(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(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)


(10) Obligation:

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

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(0) → 0
a__plus(X1, X2) → plus(X1, X2)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(a__plus(x1, x2)) = 2·x1 + x2   
POL(mark(x1)) = 2·x1   
POL(plus(x1, x2)) = 2·x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(0) → 0


(12) Obligation:

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

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(X1, X2) → plus(X1, X2)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__plus(x1, x2)) = 2 + x1 + 2·x2   
POL(mark(x1)) = 2·x1   
POL(plus(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__plus(X1, X2) → plus(X1, X2)


(14) Obligation:

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

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__plus(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
POL(plus(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(plus(X1, X2)) → a__plus(mark(X1), mark(X2))


(16) Obligation:

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

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE