Termination proof of /tmp/tmpmGEylL/Ex7

Termination of the following Term Rewriting System could be proven:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
head: {1}
2nd: {1}
take: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}

↳ CSR

  ↳ CSRInnermostProof

Context-sensitive rewrite system:
The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
head: {1}
2nd: {1}
take: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}

The CSR is orthogonal. By [10] we can switch to innermost.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

Context-sensitive rewrite system:
The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
head: {1}
2nd: {1}
take: {1, 2}
0: empty set
nil: empty set
sel: {1, 2}

Innermost Strategy.

Using Improved CS-DPs we result in the following initial Q-CSDP problem.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {from, s, head, 2nd, take, sel, HEAD, 2ND, SEL, FROM, TAKE} are replacing on all positions.
For all symbols f in {cons} we have µ(f) = {1}.
The symbols in {U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

2ND(cons(X, XS)) → HEAD(XS)
SEL(s(N), cons(X, XS)) → SEL(N, XS)

The collapsing dependency pairs are DP_c:

2ND(cons(X, XS)) → XS
SEL(s(N), cons(X, XS)) → XS

The hidden terms of R are:

from(s(X))
take(N, XS)

Every hiding context is built from:

s on positions {1}
from on positions {1}
take on positions {1, 2}

Hence, the new unhiding pairs DP_u are :

2ND(cons(X, XS)) → U(XS)
SEL(s(N), cons(X, XS)) → U(XS)
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
U(from(s(X))) → FROM(s(X))
U(take(N, XS)) → TAKE(N, XS)

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The set Q consists of the following terms:

from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 4 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ AND

              ↳ QCSDP

                ↳ QCSDPSubtermProof

              ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {from, s, head, 2nd, take, sel} are replacing on all positions.
For all symbols f in {cons} we have µ(f) = {1}.
The symbols in {U} are not replacing on any position.

The TRS P consists of the following rules:

U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The set Q consists of the following terms:

from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U(x1) = x1

Subterm Order

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ AND

              ↳ QCSDP

                ↳ QCSDPSubtermProof

                  ↳ QCSDP

                    ↳ PIsEmptyProof

              ↳ QCSDP

Q-restricted context-sensitive dependency pair problem:
The symbols in {from, s, head, 2nd, take, sel} are replacing on all positions.
For all symbols f in {cons} we have µ(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The set Q consists of the following terms:

from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ AND

              ↳ QCSDP

              ↳ QCSDP

                ↳ QCSDPSubtermProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {from, s, head, 2nd, take, sel, SEL} are replacing on all positions.
For all symbols f in {cons} we have µ(f) = {1}.

The TRS P consists of the following rules:

SEL(s(N), cons(X, XS)) → SEL(N, XS)

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The set Q consists of the following terms:

from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

SEL(s(N), cons(X, XS)) → SEL(N, XS)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2) = x1

Subterm Order

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ AND

              ↳ QCSDP

              ↳ QCSDP

                ↳ QCSDPSubtermProof

                  ↳ QCSDP

                    ↳ PIsEmptyProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {from, s, head, 2nd, take, sel} are replacing on all positions.
For all symbols f in {cons} we have µ(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(XS)
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, XS)

The set Q consists of the following terms:

from(x0)
head(cons(x0, x1))
2nd(cons(x0, x1))
take(0, x0)
take(s(x0), cons(x1, x2))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.