Termination Proof using AProVE

Term Rewriting System R:
[x, y]
app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(fact, 0) -> APP(s, 0)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
APP(fact, app(s, x)) -> APP(*, app(s, x))
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(p, app(s, x))
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(app(*, app(s, x)), y) -> APP(+, app(app(*, x), y))
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(*, x)
APP(app(+, x), app(s, y)) -> APP(s, app(app(+, x), y))
APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Rw

Dependency Pair:

APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

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(0) = 0

POL(fact) = 0

POL(*) = 0

POL(s) = 0

POL(app(x₁, x₂)) = 1 + x₂

POL(+) = 0

POL(APP(x₁, x₂)) = x₂

POL(p) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Rw

Dependency Pair:

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Rewriting Transformation

Dependency Pairs:

APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))

one new Dependency Pair is created:

APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, x))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Rw

           →DP Problem 4

             ↳Rewriting Transformation

Dependency Pairs:

APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, x))
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Rw

           →DP Problem 4

             ↳Rw

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

APP(fact, app(s, x)) -> APP(fact, x)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, x))
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

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

two new Dependency Pairs are created:

APP(app(*, app(s, 0)), y'') -> APP(app(+, 0), y'')
APP(app(*, app(s, app(s, x''))), y'') -> APP(app(+, app(app(+, app(app(*, x''), y'')), y'')), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Rw

           →DP Problem 4

             ↳Rw

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(*, app(s, app(s, x''))), y'') -> APP(app(+, app(app(+, app(app(*, x''), y'')), y'')), y'')
APP(app(*, app(s, 0)), y'') -> APP(app(+, 0), y'')
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, x))
APP(fact, app(s, x)) -> APP(fact, x)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

APP(app(*, app(s, 0)), y'') -> APP(app(+, 0), y'')

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Rw

           →DP Problem 4

             ↳Rw

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

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

one new Dependency Pair is created:

APP(fact, app(s, app(s, x''))) -> APP(fact, app(s, x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Rw

           →DP Problem 4

             ↳Rw

...

               →DP Problem 8

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(fact, app(s, app(s, x''))) -> APP(fact, app(s, x''))
APP(app(*, app(s, app(s, x''))), y'') -> APP(app(+, app(app(+, app(app(*, x''), y'')), y'')), y'')
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, x))

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

POL(0)	= 0
POL(fact)	= 0
POL(*)	= 0
POL(s)	= 0
POL(app(x₁, x₂))	= 1 + x₂
POL(+)	= 0
POL(APP(x₁, x₂))	= x₂
POL(p)	= 0