Termination Proof using AProVE

Term Rewriting System R:
[x]
app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(f, app(h, x))

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))

two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))
APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(g, app(g, x)) -> APP(h, app(g, x))

two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))

two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))

three new Dependency Pairs are created:

APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))

two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(h, app(g, x'))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(h, app(g, x'))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))

three new Dependency Pairs are created:

APP(g, app(g, app(g, x'''))) -> APP(h, app(g, x'''))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))
APP(g, app(g, app(g, app(g, x')))) -> APP(h, app(g, app(h, app(g, app(h, app(g, x'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

APP(g, app(g, app(g, app(g, x')))) -> APP(h, app(g, app(h, app(g, app(h, app(g, x'))))))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))
APP(g, app(g, app(g, x'''))) -> APP(h, app(g, x'''))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(h, app(g, x'))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)

no new Dependency Pairs are created.
The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pair:

APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

The following dependency pair can be strictly oriented:

APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g) = 0

POL(h) = 1

POL(app(x₁, x₂)) = x₁

POL(f) = 0

POL(APP(x₁, x₂)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Dependency Graph

Dependency Pair:

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Polynomial Ordering

Dependency Pairs:

APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

The following dependency pairs can be strictly oriented:

APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(g) = 1

POL(h) = 0

POL(app(x₁, x₂)) = x₁

POL(f) = 0

POL(APP(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

Termination of R successfully shown.
Duration:
0:01 minutes

POL(g)	= 0
POL(h)	= 1
POL(app(x₁, x₂))	= x₁
POL(f)	= 0
POL(APP(x₁, x₂))	= 1 + x₂