Termination Proof using AProVE

Term Rewriting System R:
[x, y]
app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, 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(app(g, x), app(c, y)) -> APP(c, app(app(g, x), y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(if, app(f, x))
APP(app(g, x), app(c, y)) -> APP(f, x)
APP(app(g, x), app(c, y)) -> APP(c, app(app(g, app(s, x)), y))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(g, app(s, x))
APP(app(g, x), app(c, y)) -> APP(s, x)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

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

Rules:

app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, 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 that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
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 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

Rules:

app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)

Rules:

app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Strategy:

innermost

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

APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

five new Dependency Pairs are created:

APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)

Rules:

app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Strategy:

innermost

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

APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))

five new Dependency Pairs are created:

APP(app(g, 0), app(c, y)) -> APP(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y))
APP(app(g, 1), app(c, y)) -> APP(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y'')))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y'')))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y'')))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y))
APP(app(g, 1), app(c, y)) -> APP(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y))
APP(app(g, 0), app(c, y)) -> APP(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))

Rules:

app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Strategy:

innermost

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

APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))

five new Dependency Pairs are created:

APP(app(g, 0), app(c, y)) -> APP(app(if, true), app(c, app(app(g, app(s, 0)), y)))
APP(app(g, 1), app(c, y)) -> APP(app(if, false), app(c, app(app(g, app(s, 1)), y)))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y''))))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 6

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y'')))))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y''))))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y)))
APP(app(g, 1), app(c, y)) -> APP(app(if, false), app(c, app(app(g, app(s, 1)), y)))
APP(app(g, 0), app(c, y)) -> APP(app(if, true), app(c, app(app(g, app(s, 0)), y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y'')))
APP(app(g, 1), app(c, y)) -> APP(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y))
APP(app(g, 0), app(c, y)) -> APP(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y'')))

Rules:

app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:18 minutes