YES Termination proof of ../tpdb/TRS/CSR/Ex3_12_Luc96a.trs
Termination of the following Term Rewriting System could be proven:

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

from(X) → cons(X, from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
sel: {1, 2}
0: empty set


↳ CSR
  ↳ Zantema-Transformation

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

from(X) → cons(X, from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
sel: {1, 2}
0: empty set

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

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof

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

from(X) → cons(X, fromInact(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, a(Z))
a(x) → x
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A(fromInact(x1)) → FROM(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → SEL(X, a(Z))

The TRS R consists of the following rules:

from(X) → cons(X, fromInact(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, a(Z))
a(x) → x
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A(fromInact(x1)) → FROM(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → SEL(X, a(Z))

The TRS R consists of the following rules:

from(X) → cons(X, fromInact(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, a(Z))
a(x) → x
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

SEL(s(X), cons(Y, Z)) → SEL(X, a(Z))

The TRS R consists of the following rules:

from(X) → cons(X, fromInact(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, a(Z))
a(x) → x
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: