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

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

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

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: Polynomial ordering with Polynomial interpretation:

POL(0) = 0

POL(g) = 0

POL(cons) = 0

POL(s) = 1

POL(h) = 0

POL(APP(x₁, x₂)) = 1 + x₁ + x₂

POL(f) = 1

POL(app(x₁, x₂)) = x₁ + x₂

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

         ↳FwdInst

           →DP Problem 4

             ↳AFS

...

               →DP Problem 5

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

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

         ↳FwdInst

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 6

             ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Nar

Dependency Pair:

APP(g, app(app(cons, 0), app(app(cons, 0), y''))) -> APP(g, app(app(cons, 0), 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), app(app(cons, 0), y''))) -> APP(g, app(app(cons, 0), 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: Polynomial ordering with Polynomial interpretation:

POL(0) = 1

POL(g) = 0

POL(cons) = 0

POL(s) = 0

POL(h) = 0

POL(APP(x₁, x₂)) = 1 + x₁ + x₂

POL(app(x₁, x₂)) = x₁ + x₂

POL(f) = 1

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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 6

             ↳AFS

...

               →DP Problem 7

                 ↳Dependency Graph

       →DP Problem 3

         ↳Nar

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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Narrowing Transformation

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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Narrowing Transformation

Dependency Pair:

APP(h, app(app(cons, 0), y'')) -> APP(h, 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

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 9

                 ↳Argument Filtering and Ordering

Dependency Pair:

APP(h, app(app(cons, 0), app(app(cons, 0), y'))) -> APP(h, 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(h, app(app(cons, 0), app(app(cons, 0), y'))) -> APP(h, app(g, 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: Polynomial ordering with Polynomial interpretation:

POL(0) = 0

POL(g) = 0

POL(cons) = 1

POL(h) = 1

POL(s) = 0

POL(APP(x₁, x₂)) = 1 + x₁ + x₂

POL(app(x₁, x₂)) = x₁ + x₂

POL(f) = 0

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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 10

                 ↳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

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