Termination Proof using AProVE

Term Rewriting System R:
[X, Y]
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACKIN(s(X), 0) -> U11(ackin(X, s(0)))
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> U22(ackin(Y, X))
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)
ACKIN(s(X), 0) -> ACKIN(X, s(0))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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)

two new Dependency Pairs are created:

ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
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:

ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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(0)) -> U21(u11(ackin(X'', s(0))), X'')

two new Dependency Pairs are created:

ACKIN(s(0), s(0)) -> U21(u11(ackout(s(s(0)))), 0)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Rewriting Transformation

Dependency Pairs:

ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))
ACKIN(s(0), s(0)) -> U21(u11(ackout(s(s(0)))), 0)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

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

ACKIN(s(0), s(0)) -> U21(u11(ackout(s(s(0)))), 0)

one new Dependency Pair is created:

ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Rewriting Transformation

Dependency Pairs:

ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

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

ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))

one new Dependency Pair is created:

ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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'')

two new Dependency Pairs are created:

ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
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 6

                 ↳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(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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), 0) -> ACKIN(X, s(0))

three new Dependency Pairs are created:

ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(0)), 0) -> ACKIN(s(0), s(0))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(0)), 0) -> ACKIN(s(0), s(0))
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), 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(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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)

eight new Dependency Pairs are created:

ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(0), s(s(0))) -> ACKIN(s(0), s(0))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳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(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(0), s(s(0))) -> ACKIN(s(0), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(0)), 0) -> ACKIN(s(0), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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)

15 new Dependency Pairs are created:

U21(ackout(s(0)), s(0)) -> ACKIN(s(0), s(0))
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
U21(ackout(0), s(s(0))) -> ACKIN(s(s(0)), 0)
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

Dependency Pairs:

U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(0))) -> ACKIN(s(s(0)), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
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(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
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(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

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

ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))

four new Dependency Pairs are created:

ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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''))

nine new Dependency Pairs are created:

ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

Dependency Pairs:

U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
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''''))))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

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

ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)

four new Dependency Pairs are created:

ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

Dependency Pairs:

U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
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''''))))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

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

U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)

four new Dependency Pairs are created:

U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Forward Instantiation Transformation

Dependency Pairs:

U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
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''''))))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

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

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

13 new Dependency Pairs are created:

U21(ackout(s(s(0))), s(X'''''')) -> ACKIN(s(X''''''), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(Y'''''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y''''''))))
U21(ackout(s(s(s(0)))), s(s(X'''''''))) -> ACKIN(s(s(X''''''')), s(s(s(0))))
U21(ackout(s(s(s(s(0))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(s(s(Y''''''')))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(s(Y'''''''))))))
U21(ackout(s(s(0))), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(s(0)))
U21(ackout(s(s(0))), s(s(s(X''''''''')))) -> ACKIN(s(s(s(X'''''''''))), s(s(0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Polynomial Ordering

Dependency Pairs:

U21(ackout(s(s(0))), s(s(s(X''''''''')))) -> ACKIN(s(s(s(X'''''''''))), s(s(0)))
U21(ackout(s(s(0))), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(s(0)))
U21(ackout(s(s(s(s(s(Y''''''')))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(s(Y'''''''))))))
U21(ackout(s(s(s(s(0))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(0)))), s(s(X'''''''))) -> ACKIN(s(s(X''''''')), s(s(s(0))))
U21(ackout(s(s(s(Y'''''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y''''''))))
U21(ackout(s(s(s(s(Y''''''))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(X'''''')) -> ACKIN(s(X''''''), s(s(0)))
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

U21(ackout(s(s(0))), s(s(s(X''''''''')))) -> ACKIN(s(s(s(X'''''''''))), s(s(0)))
U21(ackout(s(s(0))), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(s(0)))
U21(ackout(s(s(s(s(s(Y''''''')))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(s(Y'''''''))))))
U21(ackout(s(s(s(s(0))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(0)))), s(s(X'''''''))) -> ACKIN(s(s(X''''''')), s(s(s(0))))
U21(ackout(s(s(s(Y'''''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y''''''))))
U21(ackout(s(s(s(s(Y''''''))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(X'''''')) -> ACKIN(s(X''''''), s(s(0)))
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u11(ackout(X)) -> ackout(X)
u22(ackout(X)) -> ackout(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(0) = 0

POL(u11(x₁)) = 1

POL(u22(x₁)) = 1

POL(U21(x₁, x₂)) = x₁ + x₂

POL(ackin(x₁, x₂)) = 1

POL(u21(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

POL(ACKIN(x₁, x₂)) = x₁

POL(ackout(x₁)) = 1

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Dependency Graph

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(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Polynomial Ordering

Dependency Pairs:

ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
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''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
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''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(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(0) = 0

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 17

                 ↳Dependency Graph

Dependency Pair:

Rules:

ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:15 minutes

POL(0)	= 0
POL(u11(x₁))	= 1
POL(u22(x₁))	= 1
POL(U21(x₁, x₂))	= x₁ + x₂
POL(ackin(x₁, x₂))	= 1
POL(u21(x₁, x₂))	= x₁
POL(s(x₁))	= 1 + x₁
POL(ACKIN(x₁, x₂))	= x₁
POL(ackout(x₁))	= 1