Termination Proof using AProVE

Term Rewriting System R:
[X, Y]
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

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

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)

one new Dependency Pair is created:

ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

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

ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')

one new Dependency Pair is created:

ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

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

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)

two new Dependency Pairs are created:

ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

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

U21(ackout(X), Y) -> ACKIN(Y, X)

three new Dependency Pairs are created:

U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

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

ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

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

ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))

two new Dependency Pairs are created:

ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(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(s(x₁)) = 1 + x₁

POL(ACKIN(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Dependency Graph

Dependency Pair:

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes

POL(s(x₁))	= 1 + x₁
POL(ACKIN(x₁, x₂))	= 1 + x₁ + x₂