Termination Proof using AProVE

Term Rewriting System R:
[y, x]
app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(ack, 0), y) -> APP(succ, y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), y) -> APP(ack, x)
APP(app(ack, app(succ, x)), y) -> APP(succ, 0)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(ack, x)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))

Rules:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Strategy:

innermost

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

APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))

two new Dependency Pairs are created:

APP(app(ack, app(succ, x'')), app(succ, y'')) -> APP(app(ack, x''), app(app(ack, x''), app(succ, 0)))
APP(app(ack, app(succ, x'')), app(succ, app(succ, y''))) -> APP(app(ack, x''), app(app(ack, x''), app(app(ack, app(succ, x'')), y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(ack, app(succ, x'')), app(succ, app(succ, y''))) -> APP(app(ack, x''), app(app(ack, x''), app(app(ack, app(succ, x'')), y'')))
APP(app(ack, app(succ, x'')), app(succ, y'')) -> APP(app(ack, x''), app(app(ack, x''), app(succ, 0)))
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)

Rules:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Strategy:

innermost

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

APP(app(ack, app(succ, x'')), app(succ, y'')) -> APP(app(ack, x''), app(app(ack, x''), app(succ, 0)))

three new Dependency Pairs are created:

APP(app(ack, app(succ, 0)), app(succ, y'')) -> APP(app(ack, 0), app(succ, app(succ, 0)))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(app(ack, app(succ, x')), 0)))

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(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(app(ack, app(succ, x')), 0)))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, 0)), app(succ, y'')) -> APP(app(ack, 0), app(succ, app(succ, 0)))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x'')), app(succ, app(succ, y''))) -> APP(app(ack, x''), app(app(ack, x''), app(app(ack, app(succ, x'')), y'')))

Rules:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Strategy:

innermost

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

APP(app(ack, app(succ, x'')), app(succ, app(succ, y''))) -> APP(app(ack, x''), app(app(ack, x''), app(app(ack, app(succ, x'')), y'')))

four new Dependency Pairs are created:

APP(app(ack, app(succ, 0)), app(succ, app(succ, y'''))) -> APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), y''')))
APP(app(ack, app(succ, app(succ, x'))), app(succ, app(succ, y'''))) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, x''')), app(succ, app(succ, y'''))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(succ, 0))))
APP(app(ack, app(succ, x''')), app(succ, app(succ, app(succ, y')))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(app(ack, app(succ, x''')), y'))))

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(app(ack, app(succ, x''')), app(succ, app(succ, app(succ, y')))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(app(ack, app(succ, x''')), y'))))
APP(app(ack, app(succ, x''')), app(succ, app(succ, y'''))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(succ, 0))))
APP(app(ack, app(succ, app(succ, x'))), app(succ, app(succ, y'''))) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, 0)), app(succ, app(succ, y'''))) -> APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), y''')))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, 0)), app(succ, y'')) -> APP(app(ack, 0), app(succ, app(succ, 0)))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(app(ack, app(succ, x')), 0)))

Rules:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Strategy:

innermost

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

APP(app(ack, app(succ, 0)), app(succ, y'')) -> APP(app(ack, 0), app(succ, app(succ, 0)))

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Argument Filtering and Ordering

Dependency Pairs:

APP(app(ack, app(succ, x''')), app(succ, app(succ, y'''))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(succ, 0))))
APP(app(ack, app(succ, app(succ, x'))), app(succ, app(succ, y'''))) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, 0)), app(succ, app(succ, y'''))) -> APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), y''')))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(app(ack, app(succ, x')), 0)))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x''')), app(succ, app(succ, app(succ, y')))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(app(ack, app(succ, x''')), y'))))

Rules:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(ack, app(succ, x''')), app(succ, app(succ, y'''))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(succ, 0))))
APP(app(ack, app(succ, app(succ, x'))), app(succ, app(succ, y'''))) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, 0)), app(succ, app(succ, y'''))) -> APP(app(ack, 0), app(succ, app(app(ack, app(succ, 0)), y''')))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(app(ack, app(succ, x')), 0)))
APP(app(ack, app(succ, app(succ, x'))), app(succ, y'')) -> APP(app(ack, app(succ, x')), app(app(ack, x'), app(succ, 0)))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x''')), app(succ, app(succ, app(succ, y')))) -> APP(app(ack, x'''), app(app(ack, x'''), app(app(ack, x'''), app(app(ack, app(succ, x''')), y'))))

The following usable rules for innermost w.r.t. to the AFS can be oriented:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{app, 0, APP} > succ

resulting in one new DP problem.
Used Argument Filtering System:

APP(x₁, x₂) -> APP(x₁, x₂)
app(x₁, x₂) -> app(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Dependency Graph

Dependency Pair:

Rules:

app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes