YES Termination proof of ../tpdb/TRS/CSR/Ex3_3_25_Bor03.trs
Termination of the following Term Rewriting System could be proven:

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

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))

The replacement map contains the following entries:

app: {1, 2}
nil: empty set
cons: {1}
from: {1}
s: {1}
zWadr: {1, 2}
prefix: {1}


↳ CSR
  ↳ Lucas-Transformation

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

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, app(XS, YS))
from(X) → cons(X, from(s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, nil)), zWadr(XS, YS))
prefix(L) → cons(nil, zWadr(L, prefix(L)))

The replacement map contains the following entries:

app: {1, 2}
nil: empty set
cons: {1}
from: {1}
s: {1}
zWadr: {1, 2}
prefix: {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:

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X), cons(Y)) → cons(app(Y, cons(X)))
prefix(L) → cons(nil)

Q is empty.

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

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X), cons(Y)) → cons(app(Y, cons(X)))
prefix(L) → cons(nil)

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

zWadr(cons(X), cons(Y)) → cons(app(Y, cons(X)))
Used ordering:
Polynomial interpretation [25]:

POL(app(x1, x2)) = x1 + x2   
POL(cons(x1)) = 2 + x1   
POL(from(x1)) = 2 + x1   
POL(nil) = 0   
POL(prefix(x1)) = 2 + x1   
POL(zWadr(x1, x2)) = 2·x1 + x2   




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

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

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
prefix(L) → cons(nil)

Q is empty.

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

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
prefix(L) → cons(nil)

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

app(nil, YS) → YS
app(cons(X), YS) → cons(X)
from(X) → cons(X)
prefix(L) → cons(nil)
Used ordering:
Polynomial interpretation [25]:

POL(app(x1, x2)) = 2 + 2·x1 + x2   
POL(cons(x1)) = 1 + 2·x1   
POL(from(x1)) = 2 + 2·x1   
POL(nil) = 0   
POL(prefix(x1)) = 2 + x1   
POL(zWadr(x1, x2)) = x1 + 2·x2   




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

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

zWadr(nil, YS) → nil
zWadr(XS, nil) → nil

Q is empty.

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

zWadr(nil, YS) → nil
zWadr(XS, nil) → nil

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

zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
Used ordering:
Polynomial interpretation [25]:

POL(nil) = 1   
POL(zWadr(x1, x2)) = 2 + 2·x1 + 2·x2   




↳ CSR
  ↳ Lucas-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.