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

(0) Obligation:

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0) → 0
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
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__and(x1, x2)  =  a__and(x1, x2)
tt  =  tt
mark(x1)  =  x1
a__plus(x1, x2)  =  a__plus(x1, x2)
0  =  0
s(x1)  =  s(x1)
a__x(x1, x2)  =  a__x(x1, x2)
and(x1, x2)  =  and(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

[a_and2, and₂]
[a_x2, x₂] > [a_plus2, plus₂] > s₁
[a_x2, x₂] > 0

Status:

a_plus2: [1,2]
a_x2: [2,1]
plus₂: [1,2]
tt: multiset
a_and2: multiset
s₁: [1]
x₂: [2,1]
and₂: multiset
0: multiset

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

a__and(tt, X) → mark(X)
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → s(a__plus(mark(N), mark(M)))
a__x(N, 0) → 0
a__x(N, s(M)) → a__plus(a__x(mark(N), mark(M)), mark(N))

(2) Obligation:

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

mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(a__and(x₁, x₂)) = x₁ + 2·x₂   
POL(a__plus(x₁, x₂)) = x₁ + 2·x₂   
POL(a__x(x₁, x₂)) = x₁ + 2·x₂   
POL(and(x₁, x₂)) = x₁ + 2·x₂   
POL(mark(x₁)) = 2·x₁   
POL(plus(x₁, x₂)) = x₁ + 2·x₂   
POL(s(x₁)) = 1 + x₁   
POL(tt) = 0   
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(0) → 0
mark(s(X)) → s(mark(X))

(4) Obligation:

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

mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
a__and(X1, X2) → and(X1, X2)
a__plus(X1, X2) → plus(X1, X2)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__and(x₁, x₂)) = 1 + x₁ + 2·x₂   
POL(a__plus(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(a__x(x₁, x₂)) = 2·x₁ + x₂   
POL(and(x₁, x₂)) = 1 + x₁ + x₂   
POL(mark(x₁)) = 2·x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + 2·x₂   
POL(tt) = 0   
POL(x(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(and(X1, X2)) → a__and(mark(X1), X2)
a__plus(X1, X2) → plus(X1, X2)

(6) Obligation:

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

mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
a__and(X1, X2) → and(X1, X2)
a__x(X1, X2) → x(X1, X2)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__and(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__plus(x₁, x₂)) = x₁ + x₂   
POL(a__x(x₁, x₂)) = x₁ + x₂   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(mark(x₁)) = x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + x₂   
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:

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

(8) Obligation:

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

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

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a__x(x₁, x₂)) = 2·x₁ + x₂   
POL(mark(x₁)) = 2·x₁   
POL(tt) = 1   
POL(x(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(tt) → tt

(10) 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.

(11) 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)

(12) 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.

(13) 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))

(14) Obligation:

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

(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

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE