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

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(s) = 0

POL(APP(x₁, x₂)) = x₂

POL(f) = 0

POL(app(x₁, x₂)) = 1 + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

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

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

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

The following dependency pairs can be strictly oriented:

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))

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

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)))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(if) = 0

POL(c) = 0

POL(0) = 1

POL(g) = 1

POL(false) = 0

POL(1) = 0

POL(true) = 0

POL(s) = 0

POL(app(x₁, x₂)) = x₁

POL(f) = 0

POL(APP(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 4

             ↳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(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

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Remaining Obligation(s)

The following remains to be proven:
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(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

Innermost Termination of R could not be shown.
Duration:
0:03 minutes

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

POL(if)	= 0
POL(c)	= 0
POL(0)	= 1
POL(g)	= 1
POL(false)	= 0
POL(1)	= 0
POL(true)	= 0
POL(s)	= 0
POL(app(x₁, x₂))	= x₁
POL(f)	= 0
POL(APP(x₁, x₂))	= x₁