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 RisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 RisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 RisEmptyProof (⇔)
    ↳18 TRUE

(0) Obligation:

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

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U31(tt) → 0
a__U41(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
a__U11(x1, x2)  =  a__U11(x1, x2)
tt  =  tt
mark(x1)  =  x1
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
s(x1)  =  s(x1)
a__plus(x1, x2)  =  a__plus(x1, x2)
a__U31(x1)  =  x1
0  =  0
a__U41(x1, x2, x3)  =  a__U41(x1, x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
a__and(x1, x2)  =  a__and(x1, x2)
a__isNat(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
isNat(x1)  =  x1
x(x1, x2)  =  x(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U31(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
and(x1, x2)  =  and(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
[tt, 0] > [aU413, ax2, x2, U413] > [aU213, aplus2, plus2, U213] > [aU112, s1, aand2, U112, and2]

Status:
plus2: [2,1]
aand2: multiset
aU112: multiset
U413: [3,2,1]
aU213: [2,3,1]
x2: [1,2]
U112: multiset
and2: multiset
0: multiset
ax2: [1,2]
aplus2: [2,1]
aU413: [3,2,1]
tt: multiset
s1: [1]
U213: [2,3,1]

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

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U41(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)


(2) Obligation:

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

a__U31(tt) → 0
a__isNat(0) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x1, x2)) = x1 + x2   
POL(U21(x1, x2, x3)) = x1 + x2 + x3   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U11(x1, x2)) = x1 + x2   
POL(a__U21(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U31(x1)) = 1 + x1   
POL(a__U41(x1, x2, x3)) = x1 + x2 + x3   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = x1   
POL(a__plus(x1, x2)) = x1 + x2   
POL(a__x(x1, x2)) = x1 + x2   
POL(and(x1, x2)) = x1 + x2   
POL(isNat(x1)) = x1   
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__isNat(0) → tt


(4) Obligation:

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

a__U31(tt) → 0
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x1, x2)) = x1 + x2   
POL(U21(x1, x2, x3)) = 1 + 2·x1 + x2 + 2·x3   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U11(x1, x2)) = x1 + 2·x2   
POL(a__U21(x1, x2, x3)) = 2 + 2·x1 + x2 + 2·x3   
POL(a__U31(x1)) = 2 + x1   
POL(a__U41(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(a__and(x1, x2)) = 2·x1 + 2·x2   
POL(a__isNat(x1)) = 1 + 2·x1   
POL(a__plus(x1, x2)) = 2 + x1 + x2   
POL(a__x(x1, x2)) = 1 + x1 + 2·x2   
POL(and(x1, x2)) = 2·x1 + x2   
POL(isNat(x1)) = 1 + x1   
POL(mark(x1)) = 2·x1   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = 2 + 2·x1   
POL(tt) = 2   
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__U31(tt) → 0
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)


(6) Obligation:

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1, x2)) = x1 + x2   
POL(U21(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(a__U11(x1, x2)) = x1 + x2   
POL(a__U21(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2, x3)) = x1 + x2 + x3   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = 1 + x1   
POL(a__plus(x1, x2)) = x1 + x2   
POL(a__x(x1, x2)) = 1 + x1 + x2   
POL(and(x1, x2)) = x1 + x2   
POL(isNat(x1)) = x1   
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(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
a__x(X1, X2) → x(X1, X2)
a__isNat(X) → isNat(X)


(8) Obligation:

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1, x2)) = 2 + 2·x1 + x2   
POL(a__U11(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(a__and(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(and(x1, x2)) = 1 + 2·x1 + x2   
POL(mark(x1)) = 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)


(10) Obligation:

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

a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1, x2)) = 1 + x1 + x2   
POL(a__U11(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(a__and(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(and(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:

a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)


(12) Obligation:

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

(13) RisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE