YES Termination proof of ../tpdb/TRS/CSR/Ex4_4_Luc96b.trs
Termination of the following Term Rewriting System could be proven:

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

f(g(X), Y) → f(X, f(g(X), Y))

The replacement map contains the following entries:

f: {1}
g: {1}


↳ CSR
  ↳ Lucas-Transformation

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

f(g(X), Y) → f(X, f(g(X), Y))

The replacement map contains the following entries:

f: {1}
g: {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:

f(g(X)) → f(X)

Q is empty.

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

f(g(X)) → f(X)

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

f(g(X)) → f(X)
Used ordering:
Polynomial interpretation [25]:

POL(f(x1)) = 2·x1   
POL(g(x1)) = 1 + 2·x1   




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

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

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