Termination of the following Term Rewriting System could be proven:

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

f(g(X), Y) → f(X, f(g(X), Y))

The replacement map contains the following entries:

f: {1}
g: {1}


↳ CSR
  ↳ CSRInnermostProof

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

f(g(X), Y) → f(X, f(g(X), Y))

The replacement map contains the following entries:

f: {1}
g: {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:

f(g(X), Y) → f(X, f(g(X), Y))

The replacement map contains the following entries:

f: {1}
g: {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 {g} are replacing on all positions.
For all symbols f in {f, F} we have µ(f) = {1}.
The symbols in {U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DPo are:

F(g(X), Y) → F(X, f(g(X), Y))


The hidden terms of R are:

f(g(X), Y)

Every hiding context is built from:

g on positions {1}
f on positions {1}

Hence, the new unhiding pairs DPu are :

U(g(x_0)) → U(x_0)
U(f(x_0, x_1)) → U(x_0)
U(f(g(X), Y)) → F(g(X), Y)

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)


The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 1 less node.


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

Q-restricted context-sensitive dependency pair problem:
The symbols in {g} are replacing on all positions.
For all symbols f in {f, F} we have µ(f) = {1}.

The TRS P consists of the following rules:

F(g(X), Y) → F(X, f(g(X), Y))

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)


We use the subterm processor [20].


The following pairs can be oriented strictly and are deleted.


F(g(X), Y) → F(X, f(g(X), Y))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
F(x1, x2)  =  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 {g} are replacing on all positions.
For all symbols f in {f} we have µ(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)


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 {g} are replacing on all positions.
For all symbols f in {f} we have µ(f) = {1}.
The symbols in {U} are not replacing on any position.

The TRS P consists of the following rules:

U(g(x_0)) → U(x_0)
U(f(x_0, x_1)) → U(x_0)

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)


We use the subterm processor [20].


The following pairs can be oriented strictly and are deleted.


U(g(x_0)) → U(x_0)
U(f(x_0, x_1)) → 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
            ↳ AND
              ↳ QCSDP
              ↳ QCSDP
                ↳ QCSDPSubtermProof
                  ↳ QCSDP
                    ↳ PIsEmptyProof

Q-restricted context-sensitive dependency pair problem:
The symbols in {g} are replacing on all positions.
For all symbols f in {f} we have µ(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)


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