Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) -> app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) -> app(app(+, x), app(app(+, y), z))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(+, app(app(*, x), y))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, app(app(+, y), z)), x) -> APP(+, app(app(*, x), y))
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), y)
APP(app(*, app(app(+, y), z)), x) -> APP(*, x)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), z)
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, y), z)
APP(app(*, app(app(*, x), y)), z) -> APP(*, y)
APP(app(+, app(app(+, x), y)), z) -> APP(app(+, x), app(app(+, y), z))
APP(app(+, app(app(+, x), y)), z) -> APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) -> APP(+, y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

Dependency Pairs:

APP(app(+, app(app(+, x), y)), z) -> APP(app(+, y), z)
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, y), z)
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))

Rules:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) -> app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) -> app(app(+, x), app(app(+, y), z))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(*, app(app(+, y), z)), x) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))

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

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) -> app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) -> app(app(+, x), app(app(+, y), z))

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

* > +

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

APP(x₁, x₂) -> x₁
app(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 2

             ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

Rules:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) -> app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) -> app(app(+, x), app(app(+, y), z))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes