YES Termination proof of ../tpdb/TRS/CSR/Ex5_Zan97.trs
Termination of the following Term Rewriting System could be proven:

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

f(X) → if(X, c, f(true))
if(true, X, Y) → X
if(false, X, Y) → Y

The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set
false: empty set


↳ CSR
  ↳ Zantema-Transformation

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

f(X) → if(X, c, f(true))
if(true, X, Y) → X
if(false, X, Y) → Y

The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set
false: empty set

We applied the Zantema [34] to transform the context-sensitive TRS to an usual TRS.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

f(X) → if(X, c, fInact(true))
if(true, X, Y) → X
if(false, X, Y) → a(Y)
a(x) → x
f(x1) → fInact(x1)
a(fInact(x1)) → f(x1)

Q is empty.

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

f(X) → if(X, c, fInact(true))
if(true, X, Y) → X
if(false, X, Y) → a(Y)
a(x) → x
f(x1) → fInact(x1)
a(fInact(x1)) → f(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(x) → x
a(fInact(x1)) → f(x1)
Used ordering:
Polynomial interpretation [25]:

POL(a(x1)) = 1 + x1   
POL(c) = 0   
POL(f(x1)) = 2·x1   
POL(fInact(x1)) = 2·x1   
POL(false) = 1   
POL(if(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(true) = 0   




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

f(X) → if(X, c, fInact(true))
if(true, X, Y) → X
if(false, X, Y) → a(Y)
f(x1) → fInact(x1)

Q is empty.

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

f(X) → if(X, c, fInact(true))
if(true, X, Y) → X
if(false, X, Y) → a(Y)
f(x1) → fInact(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:

f(X) → if(X, c, fInact(true))
if(false, X, Y) → a(Y)
f(x1) → fInact(x1)
Used ordering:
Polynomial interpretation [25]:

POL(a(x1)) = 2 + x1   
POL(c) = 0   
POL(f(x1)) = 2 + 2·x1   
POL(fInact(x1)) = 1 + 2·x1   
POL(false) = 2   
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(true) = 0   




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof

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

if(true, X, Y) → X

Q is empty.

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

if(true, X, Y) → 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:

if(true, X, Y) → X
Used ordering:
Polynomial interpretation [25]:

POL(if(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(true) = 0   




↳ CSR
  ↳ Zantema-Transformation
    ↳ 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.