Termination Proof using AProVE

Term Rewriting System R:
[x, y]
app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

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

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

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

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

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

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

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

APP(D, app(app(*, x), y)) -> APP(app(*, y), app(D, x))

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

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

APP(D, app(app(-, x), y)) -> APP(app(-, app(D, x)), app(D, y))

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(D, app(app(-, x), y)) -> APP(D, y)
APP(D, app(app(-, x), y)) -> APP(D, 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(*) = 0

POL(D) = 0

POL(-) = 1

POL(APP(x₁, x₂)) = 1 + x₂

POL(app(x₁, x₂)) = x₁ + x₂

POL(+) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(D, app(app(*, x), y)) -> APP(D, y)
APP(D, app(app(*, x), y)) -> APP(D, 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(*) = 1

POL(D) = 0

POL(APP(x₁, x₂)) = 1 + x₂

POL(app(x₁, x₂)) = x₁ + x₂

POL(+) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(D, app(app(+, x), y)) -> APP(D, y)
APP(D, app(app(+, x), y)) -> APP(D, 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(D) = 0

POL(APP(x₁, x₂)) = 1 + x₂

POL(app(x₁, x₂)) = x₁ + x₂

POL(+) = 1

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Dependency Graph

Dependency Pair:

Rules:

app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:04 minutes

POL(*)	= 0
POL(D)	= 0
POL(-)	= 1
POL(APP(x₁, x₂))	= 1 + x₂
POL(app(x₁, x₂))	= x₁ + x₂
POL(+)	= 0