Termination Proof using AProVE

Term Rewriting System R:
[x, y]
app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(f, app(s, x)) -> APP(f, x)
APP(g, app(app(cons, 0), y)) -> APP(g, y)
APP(h, app(app(cons, x), y)) -> APP(h, app(g, app(app(cons, x), y)))
APP(h, app(app(cons, x), y)) -> APP(g, app(app(cons, x), y))

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

Dependency Pair:

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

Rules:

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

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

         ↳AFS

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

Dependency Pair:

Rules:

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

Dependency Pair:

APP(g, app(app(cons, 0), y)) -> APP(g, y)

Rules:

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(g, app(app(cons, 0), y)) -> APP(g, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

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

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

Dependency Pair:

Rules:

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

Dependency Pair:

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

Rules:

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

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

cons > {g, s}

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

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 6

             ↳Dependency Graph

Dependency Pair:

Rules:

app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes