YES Termination proof of ../tpdb/TRS/CSR_Maude/palindrome/PALINDROME_nosorts-noand.trs
Termination of the following Term Rewriting System could be proven:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(I, __(P, I))) → U11(tt)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}


↳ CSR
  ↳ Lucas-Transformation

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(I, __(P, I))) → U11(tt)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}

We applied the Lucas [26] to transform the context-sensitive TRS to an usual TRS.

↳ CSR
  ↳ Lucas-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(I, __(P, I))) → U11(tt)

Q is empty.

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → U12(tt)
U12(tt) → tt
isNePal(__(I, __(P, I))) → U11(tt)

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

__(X, nil) → X
__(nil, X) → X
isNePal(__(I, __(P, I))) → U11(tt)
Used ordering:
Polynomial interpretation [25]:

POL(U11(x1)) = x1   
POL(U12(x1)) = 2·x1   
POL(__(x1, x2)) = x1 + x2   
POL(isNePal(x1)) = 1 + 2·x1   
POL(nil) = 1   
POL(tt) = 0   




↳ CSR
  ↳ Lucas-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → U12(tt)
U12(tt) → tt

Q is empty.

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

__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → U12(tt)
U12(tt) → tt

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

__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → U12(tt)
U12(tt) → tt
Used ordering:
Polynomial interpretation [25]:

POL(U11(x1)) = 2 + 2·x1   
POL(U12(x1)) = 1 + 2·x1   
POL(__(x1, x2)) = 1 + 2·x1 + x2   
POL(tt) = 2   




↳ CSR
  ↳ Lucas-Transformation
    ↳ 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.