Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a(b(c(x1))) → c(b(a(x1)))
C(B(A(x1))) → A(B(C(x1)))
b(a(C(x1))) → C(a(b(x1)))
c(A(B(x1))) → B(A(c(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))
a(A(x1)) → x1
A(a(x1)) → x1
b(B(x1)) → x1
B(b(x1)) → x1
c(C(x1)) → x1
C(c(x1)) → x1

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

a(b(c(x1))) → c(b(a(x1)))
C(B(A(x1))) → A(B(C(x1)))
b(a(C(x1))) → C(a(b(x1)))
c(A(B(x1))) → B(A(c(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))
a(A(x1)) → x1
A(a(x1)) → x1
b(B(x1)) → x1
B(b(x1)) → x1
c(C(x1)) → x1
C(c(x1)) → x1

Q is empty.

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

a(b(c(x1))) → c(b(a(x1)))
C(B(A(x1))) → A(B(C(x1)))
b(a(C(x1))) → C(a(b(x1)))
c(A(B(x1))) → B(A(c(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))
a(A(x1)) → x1
A(a(x1)) → x1
b(B(x1)) → x1
B(b(x1)) → x1
c(C(x1)) → x1
C(c(x1)) → x1

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a(b(c(x1))) → c(b(a(x1)))
c(C(x1)) → x1
C(c(x1)) → x1
Used ordering:
Polynomial interpretation [25]:

POL(A(x1)) = x1   
POL(B(x1)) = x1   
POL(C(x1)) = x1   
POL(a(x1)) = 2·x1   
POL(b(x1)) = x1   
POL(c(x1)) = 1 + 2·x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

C(B(A(x1))) → A(B(C(x1)))
b(a(C(x1))) → C(a(b(x1)))
c(A(B(x1))) → B(A(c(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))
a(A(x1)) → x1
A(a(x1)) → x1
b(B(x1)) → x1
B(b(x1)) → x1

Q is empty.

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

C(B(A(x1))) → A(B(C(x1)))
b(a(C(x1))) → C(a(b(x1)))
c(A(B(x1))) → B(A(c(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))
a(A(x1)) → x1
A(a(x1)) → x1
b(B(x1)) → x1
B(b(x1)) → x1

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

b(a(C(x1))) → C(a(b(x1)))
c(A(B(x1))) → B(A(c(x1)))
a(A(x1)) → x1
A(a(x1)) → x1
b(B(x1)) → x1
B(b(x1)) → x1
Used ordering:
Polynomial interpretation [25]:

POL(A(x1)) = 1 + 2·x1   
POL(B(x1)) = x1   
POL(C(x1)) = x1   
POL(a(x1)) = 2 + 2·x1   
POL(b(x1)) = 1 + 2·x1   
POL(c(x1)) = 2·x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

C(B(A(x1))) → A(B(C(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))

Q is empty.

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

C(B(A(x1))) → A(B(C(x1)))
A(c(b(x1))) → b(c(A(x1)))
B(C(a(x1))) → a(C(B(x1)))

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

C(B(A(x1))) → A(B(C(x1)))
B(C(a(x1))) → a(C(B(x1)))
Used ordering:
Polynomial interpretation [25]:

POL(A(x1)) = 2 + x1   
POL(B(x1)) = 2·x1   
POL(C(x1)) = 2 + 2·x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = x1   
POL(c(x1)) = x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof

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

A(c(b(x1))) → b(c(A(x1)))

Q is empty.

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

A(c(b(x1))) → b(c(A(x1)))

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

A(c(b(x1))) → b(c(A(x1)))
Used ordering:
Polynomial interpretation [25]:

POL(A(x1)) = 2·x1   
POL(b(x1)) = 2 + x1   
POL(c(x1)) = 1 + 2·x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RisEmptyProof

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

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