Termination Proof using AProVE

Term Rewriting System R:
[x, y]
top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP1(free(x), y) -> CHECK(new(x))
TOP1(free(x), y) -> NEW(x)
TOP1(free(x), y) -> TOP2(new(x), check(y))
TOP1(free(x), y) -> CHECK(y)
TOP1(free(x), y) -> TOP2(check(x), new(y))
TOP1(free(x), y) -> CHECK(x)
TOP1(free(x), y) -> NEW(y)
TOP1(free(x), y) -> TOP2(x, check(new(y)))
TOP1(free(x), y) -> CHECK(new(y))
TOP2(x, free(y)) -> TOP1(check(new(x)), y)
TOP2(x, free(y)) -> CHECK(new(x))
TOP2(x, free(y)) -> NEW(x)
TOP2(x, free(y)) -> TOP1(new(x), check(y))
TOP2(x, free(y)) -> CHECK(y)
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP2(x, free(y)) -> CHECK(x)
TOP2(x, free(y)) -> NEW(y)
TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP2(x, free(y)) -> CHECK(new(y))
NEW(free(x)) -> NEW(x)
OLD(free(x)) -> OLD(x)
CHECK(free(x)) -> CHECK(x)
CHECK(new(x)) -> NEW(check(x))
CHECK(new(x)) -> CHECK(x)
CHECK(old(x)) -> OLD(check(x))
CHECK(old(x)) -> CHECK(x)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

NEW(free(x)) -> NEW(x)

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

The following dependency pair can be strictly oriented:

NEW(free(x)) -> NEW(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(NEW(x₁)) = x₁

POL(free(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

OLD(free(x)) -> OLD(x)

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

The following dependency pair can be strictly oriented:

OLD(free(x)) -> OLD(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(free(x₁)) = 1 + x₁

POL(OLD(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 6

             ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

CHECK(old(x)) -> CHECK(x)
CHECK(new(x)) -> CHECK(x)
CHECK(free(x)) -> CHECK(x)

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

The following dependency pair can be strictly oriented:

CHECK(old(x)) -> CHECK(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(old(x₁)) = 1 + x₁

POL(free(x₁)) = x₁

POL(CHECK(x₁)) = x₁

POL(new(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 7

             ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pairs:

CHECK(new(x)) -> CHECK(x)
CHECK(free(x)) -> CHECK(x)

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

The following dependency pair can be strictly oriented:

CHECK(new(x)) -> CHECK(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(free(x₁)) = x₁

POL(CHECK(x₁)) = x₁

POL(new(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 7

             ↳Polo

...

               →DP Problem 8

                 ↳Polynomial Ordering

       →DP Problem 4

         ↳Nar

Dependency Pair:

CHECK(free(x)) -> CHECK(x)

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

The following dependency pair can be strictly oriented:

CHECK(free(x)) -> CHECK(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(free(x₁)) = 1 + x₁

POL(CHECK(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 7

             ↳Polo

...

               →DP Problem 9

                 ↳Dependency Graph

       →DP Problem 4

         ↳Nar

Dependency Pair:

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Narrowing Transformation

Dependency Pairs:

TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP1(free(x), y) -> TOP2(x, check(new(y)))
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP1(free(x), y) -> TOP2(check(x), new(y))
TOP2(x, free(y)) -> TOP1(new(x), check(y))
TOP1(free(x), y) -> TOP2(new(x), check(y))
TOP2(x, free(y)) -> TOP1(check(new(x)), y)
TOP1(free(x), y) -> TOP2(check(new(x)), y)

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

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

TOP1(free(x), y) -> TOP2(new(x), check(y))

six new Dependency Pairs are created:

TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 10

             ↳Narrowing Transformation

Dependency Pairs:

TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP1(free(x), y) -> TOP2(x, check(new(y)))
TOP2(x, free(y)) -> TOP1(new(x), check(y))
TOP1(free(x), y) -> TOP2(check(x), new(y))
TOP2(x, free(y)) -> TOP1(check(new(x)), y)
TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP2(x, free(y)) -> TOP1(x, check(new(y)))

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

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

TOP1(free(x), y) -> TOP2(check(x), new(y))

six new Dependency Pairs are created:

TOP1(free(free(x'')), y) -> TOP2(free(check(x'')), new(y))
TOP1(free(new(x'')), y) -> TOP2(new(check(x'')), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(check(x'')), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(x''), new(y))
TOP1(free(x), free(x'')) -> TOP2(check(x), free(new(x'')))
TOP1(free(x), serve) -> TOP2(check(x), free(serve))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

TOP1(free(x), serve) -> TOP2(check(x), free(serve))
TOP1(free(x), free(x'')) -> TOP2(check(x), free(new(x'')))
TOP1(free(old(x'')), y) -> TOP2(old(x''), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(check(x'')), new(y))
TOP1(free(new(x'')), y) -> TOP2(new(check(x'')), new(y))
TOP1(free(free(x'')), y) -> TOP2(free(check(x'')), new(y))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP1(free(x), y) -> TOP2(x, check(new(y)))
TOP2(x, free(y)) -> TOP1(new(x), check(y))
TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP2(x, free(y)) -> TOP1(check(new(x)), y)
TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

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

TOP1(free(x), y) -> TOP2(x, check(new(y)))

three new Dependency Pairs are created:

TOP1(free(x), y') -> TOP2(x, new(check(y')))
TOP1(free(x), free(x'')) -> TOP2(x, check(free(new(x''))))
TOP1(free(x), serve) -> TOP2(x, check(free(serve)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

TOP1(free(x), serve) -> TOP2(x, check(free(serve)))
TOP1(free(x), free(x'')) -> TOP2(x, check(free(new(x''))))
TOP1(free(x), y') -> TOP2(x, new(check(y')))
TOP1(free(x), free(x'')) -> TOP2(check(x), free(new(x'')))
TOP1(free(old(x'')), y) -> TOP2(old(x''), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(check(x'')), new(y))
TOP1(free(new(x'')), y) -> TOP2(new(check(x'')), new(y))
TOP1(free(free(x'')), y) -> TOP2(free(check(x'')), new(y))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP2(x, free(y)) -> TOP1(new(x), check(y))
TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP2(x, free(y)) -> TOP1(check(new(x)), y)
TOP1(free(x), serve) -> TOP2(check(x), free(serve))

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

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

TOP2(x, free(y)) -> TOP1(check(new(x)), y)

three new Dependency Pairs are created:

TOP2(x'', free(y)) -> TOP1(new(check(x'')), y)
TOP2(free(x''), free(y)) -> TOP1(check(free(new(x''))), y)
TOP2(serve, free(y)) -> TOP1(check(free(serve)), y)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

TOP1(free(x), free(x'')) -> TOP2(x, check(free(new(x''))))
TOP1(free(x), y') -> TOP2(x, new(check(y')))
TOP1(free(x), serve) -> TOP2(check(x), free(serve))
TOP1(free(x), free(x'')) -> TOP2(check(x), free(new(x'')))
TOP1(free(old(x'')), y) -> TOP2(old(x''), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(check(x'')), new(y))
TOP1(free(new(x'')), y) -> TOP2(new(check(x'')), new(y))
TOP1(free(free(x'')), y) -> TOP2(free(check(x'')), new(y))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP2(serve, free(y)) -> TOP1(check(free(serve)), y)
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP2(free(x''), free(y)) -> TOP1(check(free(new(x''))), y)
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP2(x'', free(y)) -> TOP1(new(check(x'')), y)
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP2(x, free(y)) -> TOP1(new(x), check(y))
TOP1(free(x), serve) -> TOP2(x, check(free(serve)))

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

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

TOP2(x, free(y)) -> TOP1(new(x), check(y))

six new Dependency Pairs are created:

TOP2(free(x''), free(y)) -> TOP1(free(new(x'')), check(y))
TOP2(serve, free(y)) -> TOP1(free(serve), check(y))
TOP2(x, free(free(x''))) -> TOP1(new(x), free(check(x'')))
TOP2(x, free(new(x''))) -> TOP1(new(x), new(check(x'')))
TOP2(x, free(old(x''))) -> TOP1(new(x), old(check(x'')))
TOP2(x, free(old(x''))) -> TOP1(new(x), old(x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

TOP1(free(x), serve) -> TOP2(x, check(free(serve)))
TOP1(free(x), y') -> TOP2(x, new(check(y')))
TOP1(free(x), serve) -> TOP2(check(x), free(serve))
TOP1(free(x), free(x'')) -> TOP2(check(x), free(new(x'')))
TOP1(free(old(x'')), y) -> TOP2(old(x''), new(y))
TOP2(x, free(old(x''))) -> TOP1(new(x), old(x''))
TOP1(free(old(x'')), y) -> TOP2(old(check(x'')), new(y))
TOP2(x, free(old(x''))) -> TOP1(new(x), old(check(x'')))
TOP1(free(new(x'')), y) -> TOP2(new(check(x'')), new(y))
TOP2(x, free(new(x''))) -> TOP1(new(x), new(check(x'')))
TOP1(free(free(x'')), y) -> TOP2(free(check(x'')), new(y))
TOP2(x, free(free(x''))) -> TOP1(new(x), free(check(x'')))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))
TOP2(serve, free(y)) -> TOP1(free(serve), check(y))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP2(free(x''), free(y)) -> TOP1(free(new(x'')), check(y))
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP2(serve, free(y)) -> TOP1(check(free(serve)), y)
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP2(free(x''), free(y)) -> TOP1(check(free(new(x''))), y)
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP2(x'', free(y)) -> TOP1(new(check(x'')), y)
TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP2(x, free(y)) -> TOP1(check(x), new(y))
TOP1(free(x), free(x'')) -> TOP2(x, check(free(new(x''))))

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

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

TOP2(x, free(y)) -> TOP1(check(x), new(y))

six new Dependency Pairs are created:

TOP2(free(x''), free(y)) -> TOP1(free(check(x'')), new(y))
TOP2(new(x''), free(y)) -> TOP1(new(check(x'')), new(y))
TOP2(old(x''), free(y)) -> TOP1(old(check(x'')), new(y))
TOP2(old(x''), free(y)) -> TOP1(old(x''), new(y))
TOP2(x, free(free(x''))) -> TOP1(check(x), free(new(x'')))
TOP2(x, free(serve)) -> TOP1(check(x), free(serve))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 15

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

TOP2(x, free(serve)) -> TOP1(check(x), free(serve))
TOP1(free(x), free(x'')) -> TOP2(x, check(free(new(x''))))
TOP2(x, free(free(x''))) -> TOP1(check(x), free(new(x'')))
TOP1(free(x), y') -> TOP2(x, new(check(y')))
TOP2(old(x''), free(y)) -> TOP1(old(x''), new(y))
TOP1(free(x), serve) -> TOP2(check(x), free(serve))
TOP2(old(x''), free(y)) -> TOP1(old(check(x'')), new(y))
TOP1(free(x), free(x'')) -> TOP2(check(x), free(new(x'')))
TOP2(new(x''), free(y)) -> TOP1(new(check(x'')), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(x''), new(y))
TOP2(free(x''), free(y)) -> TOP1(free(check(x'')), new(y))
TOP1(free(old(x'')), y) -> TOP2(old(check(x'')), new(y))
TOP2(x, free(old(x''))) -> TOP1(new(x), old(x''))
TOP1(free(new(x'')), y) -> TOP2(new(check(x'')), new(y))
TOP2(x, free(old(x''))) -> TOP1(new(x), old(check(x'')))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(x''))
TOP2(x, free(new(x''))) -> TOP1(new(x), new(check(x'')))
TOP1(free(free(x'')), y) -> TOP2(free(check(x'')), new(y))
TOP2(x, free(free(x''))) -> TOP1(new(x), free(check(x'')))
TOP1(free(x), old(x'')) -> TOP2(new(x), old(check(x'')))
TOP2(serve, free(y)) -> TOP1(free(serve), check(y))
TOP1(free(x), new(x'')) -> TOP2(new(x), new(check(x'')))
TOP2(serve, free(y)) -> TOP1(check(free(serve)), y)
TOP1(free(x), free(x'')) -> TOP2(new(x), free(check(x'')))
TOP2(free(x''), free(y)) -> TOP1(free(new(x'')), check(y))
TOP1(free(serve), y) -> TOP2(free(serve), check(y))
TOP2(free(x''), free(y)) -> TOP1(check(free(new(x''))), y)
TOP1(free(free(x'')), y) -> TOP2(free(new(x'')), check(y))
TOP2(x'', free(y)) -> TOP1(new(check(x'')), y)
TOP1(free(x), y) -> TOP2(check(new(x)), y)
TOP2(x, free(y)) -> TOP1(x, check(new(y)))
TOP1(free(x), serve) -> TOP2(x, check(free(serve)))

Rules:

top1(free(x), y) -> top2(check(new(x)), y)
top1(free(x), y) -> top2(new(x), check(y))
top1(free(x), y) -> top2(check(x), new(y))
top1(free(x), y) -> top2(x, check(new(y)))
top2(x, free(y)) -> top1(check(new(x)), y)
top2(x, free(y)) -> top1(new(x), check(y))
top2(x, free(y)) -> top1(check(x), new(y))
top2(x, free(y)) -> top1(x, check(new(y)))
new(free(x)) -> free(new(x))
new(serve) -> free(serve)
old(free(x)) -> free(old(x))
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Termination of R could not be shown.
Duration:
0:09 minutes

POL(old(x₁))	= 1 + x₁
POL(free(x₁))	= x₁
POL(CHECK(x₁))	= x₁
POL(new(x₁))	= x₁