Termination of the following Term Rewriting System could be proven:

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

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons: {1}
from: {1}
s: {1}


↳ CSR
  ↳ CSRInnermostProof

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

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons: {1}
from: {1}
s: {1}

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:

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons: {1}
from: {1}
s: {1}

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 {2nd, from, s, 2ND, FROM} 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 collapsing dependency pairs are DPc:

2ND(cons(X, cons(Y, Z))) → Y


The hidden terms of R are:

from(s(X))

Every hiding context is built from:

s on positions {1}
from on positions {1}

Hence, the new unhiding pairs DPu are :

2ND(cons(X, cons(Y, Z))) → U(Y)
U(s(x_0)) → U(x_0)
U(from(x_0)) → U(x_0)
U(from(s(X))) → FROM(s(X))

The TRS R consists of the following rules:

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

2nd(cons(x0, cons(x1, x2)))
from(x0)


The approximation of the Context-Sensitive Dependency Graph contains 1 SCC with 2 less nodes.


↳ CSR
  ↳ CSRInnermostProof
    ↳ CSR
      ↳ CSDependencyPairsProof
        ↳ QCSDP
          ↳ QCSDependencyGraphProof
            ↳ QCSDP
              ↳ QCSDPSubtermProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {2nd, from, s} 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)

The TRS R consists of the following rules:

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

2nd(cons(x0, cons(x1, x2)))
from(x0)


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)
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
            ↳ QCSDP
              ↳ QCSDPSubtermProof
                ↳ QCSDP
                  ↳ PIsEmptyProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {2nd, from, s} 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:

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

2nd(cons(x0, cons(x1, x2)))
from(x0)


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