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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACKIN(s(m), 0) -> U11(ackin(m, s(0)))
ACKIN(s(m), 0) -> ACKIN(m, s(0))
ACKIN(s(m), s(n)) -> U21(ackin(s(m), n), m)
ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
U21(ackout(n), m) -> U22(ackin(m, n))
U21(ackout(n), m) -> ACKIN(m, n)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
U21(ackout(n), m) -> ACKIN(m, n)
ACKIN(s(m), s(n)) -> U21(ackin(s(m), n), m)
ACKIN(s(m), 0) -> ACKIN(m, s(0))

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(m), s(n)) -> U21(ackin(s(m), n), m)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(m''), s(0)) -> U21(u11(ackin(m'', s(0))), m'')
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

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:

`   R`
`     ↳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(m''), s(s(n''))) -> U21(u21(ackin(s(m''), n''), m''), m'')
ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
ACKIN(s(m), 0) -> ACKIN(m, s(0))
U21(ackout(n), m) -> ACKIN(m, n)
ACKIN(s(s(m')), s(0)) -> U21(u11(u21(ackin(s(m'), 0), m')), s(m'))

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(s(m')), s(0)) -> U21(u11(u21(ackin(s(m'), 0), m')), s(m'))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(m''), s(s(n''))) -> U21(u21(ackin(s(m''), n''), m''), m'')
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(m), 0) -> ACKIN(m, s(0))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
eight new Dependency Pairs are created:

ACKIN(s(m''), s(s(n''))) -> ACKIN(s(m''), s(n''))
ACKIN(s(0), s(s(0))) -> ACKIN(s(0), s(0))
ACKIN(s(s(m''')), s(s(0))) -> ACKIN(s(s(m''')), s(0))
ACKIN(s(m'), s(s(s(0)))) -> ACKIN(s(m'), s(s(0)))
ACKIN(s(m'), s(s(s(s(n'''))))) -> ACKIN(s(m'), s(s(s(n'''))))
ACKIN(s(s(m'''')), s(0)) -> ACKIN(s(s(m'''')), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(s(m'''''))), s(0)) -> ACKIN(s(s(s(m'''''))), 0)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

U21(ackout(n), m) -> ACKIN(m, n)
15 new Dependency Pairs are created:

U21(ackout(s(0)), s(0)) -> ACKIN(s(0), s(0))
U21(ackout(s(0)), s(s(m'''))) -> ACKIN(s(s(m''')), s(0))
U21(ackout(s(s(0))), s(m''''')) -> ACKIN(s(m'''''), s(s(0)))
U21(ackout(s(s(s(n''')))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(n'''))))
U21(ackout(0), s(s(m''''))) -> ACKIN(s(s(m'''')), 0)
U21(ackout(0), s(s(0))) -> ACKIN(s(s(0)), 0)
U21(ackout(0), s(s(s(m''''')))) -> ACKIN(s(s(s(m'''''))), 0)
U21(ackout(s(s(n''''))), s(m'''')) -> ACKIN(s(m''''), s(s(n'''')))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(s(s(0))), s(s(m'''''))) -> ACKIN(s(s(m''''')), s(s(0)))
U21(ackout(s(s(s(0)))), s(m''')) -> ACKIN(s(m'''), s(s(s(0))))
U21(ackout(s(s(s(s(n'''''))))), s(m''')) -> ACKIN(s(m'''), s(s(s(s(n''''')))))
U21(ackout(s(0)), s(s(m''''''))) -> ACKIN(s(s(m'''''')), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(s(m''''''')))) -> ACKIN(s(s(s(m'''''''))), s(0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(s(m'')), 0) -> ACKIN(s(m''), s(0))
four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(m''), s(s(n''))) -> ACKIN(s(m''), s(n''))
nine new Dependency Pairs are created:

ACKIN(s(s(m'''')), s(s(0))) -> ACKIN(s(s(m'''')), s(0))
ACKIN(s(m'''), s(s(s(0)))) -> ACKIN(s(m'''), s(s(0)))
ACKIN(s(m'''), s(s(s(s(n''''))))) -> ACKIN(s(m'''), s(s(s(n''''))))
ACKIN(s(m''''), s(s(s(n'''')))) -> ACKIN(s(m''''), s(s(n'''')))
ACKIN(s(s(m''''')), s(s(s(0)))) -> ACKIN(s(s(m''''')), s(s(0)))
ACKIN(s(m''''), s(s(s(s(0))))) -> ACKIN(s(m''''), s(s(s(0))))
ACKIN(s(m''''), s(s(s(s(s(n''''')))))) -> ACKIN(s(m''''), s(s(s(s(n''''')))))
ACKIN(s(s(m'''''')), s(s(0))) -> ACKIN(s(s(m'''''')), s(0))
ACKIN(s(s(s(m'''''''))), s(s(0))) -> ACKIN(s(s(s(m'''''''))), s(0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 11`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

ACKIN(s(s(m'''')), s(0)) -> ACKIN(s(s(m'''')), 0)
four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳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(m''''''))) -> ACKIN(s(s(m'''''')), s(0))
U21(ackout(s(s(s(s(n'''''))))), s(m''')) -> ACKIN(s(m'''), s(s(s(s(n''''')))))
U21(ackout(s(s(s(0)))), s(m''')) -> ACKIN(s(m'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(m'''''))) -> ACKIN(s(s(m''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(m''''), s(s(s(s(s(n''''')))))) -> ACKIN(s(m''''), s(s(s(s(n''''')))))
ACKIN(s(m''''), s(s(s(s(0))))) -> ACKIN(s(m''''), s(s(s(0))))
ACKIN(s(s(m''''')), s(s(s(0)))) -> ACKIN(s(s(m''''')), s(s(0)))
ACKIN(s(m''''), s(s(s(n'''')))) -> ACKIN(s(m''''), s(s(n'''')))
ACKIN(s(m'''), s(s(s(s(n''''))))) -> ACKIN(s(m'''), s(s(s(n''''))))
ACKIN(s(m'''), s(s(s(0)))) -> ACKIN(s(m'''), s(s(0)))
ACKIN(s(m'), s(s(s(s(n'''))))) -> ACKIN(s(m'), s(s(s(n'''))))
ACKIN(s(s(s(m'''''''))), s(s(0))) -> ACKIN(s(s(s(m'''''''))), s(0))
ACKIN(s(s(m'''''')), s(s(0))) -> ACKIN(s(s(m'''''')), s(0))
ACKIN(s(s(m'''')), s(s(0))) -> ACKIN(s(s(m'''')), s(0))
ACKIN(s(m'), s(s(s(0)))) -> ACKIN(s(m'), s(s(0)))
ACKIN(s(s(m''')), s(s(0))) -> ACKIN(s(s(m''')), s(0))
U21(ackout(s(s(n''''))), s(m'''')) -> ACKIN(s(m''''), s(s(n'''')))
U21(ackout(0), s(s(s(m''''')))) -> ACKIN(s(s(s(m'''''))), 0)
U21(ackout(0), s(s(m''''))) -> ACKIN(s(s(m'''')), 0)
ACKIN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ackin(s(m'''), n'), m'''), m'''), m''')
U21(ackout(s(s(s(n''')))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(n'''))))
ACKIN(s(m'''), s(s(0))) -> U21(u21(u11(ackin(m''', s(0))), m'''), m''')
U21(ackout(s(s(0))), s(m''''')) -> ACKIN(s(m'''''), s(s(0)))
ACKIN(s(s(s(s(m''''''')))), 0) -> ACKIN(s(s(s(m'''''''))), s(0))
ACKIN(s(s(s(s(m''''''''')))), s(0)) -> ACKIN(s(s(s(s(m''''''''')))), 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(m''''''))), 0) -> ACKIN(s(s(m'''''')), s(0))
ACKIN(s(s(s(m''''''''))), s(0)) -> ACKIN(s(s(s(m''''''''))), 0)
ACKIN(s(s(s(m''''))), 0) -> ACKIN(s(s(m'''')), s(0))
ACKIN(s(s(s(m''''''))), s(0)) -> ACKIN(s(s(s(m''''''))), 0)
ACKIN(s(s(s(m'''))), 0) -> ACKIN(s(s(m''')), s(0))
ACKIN(s(s(s(m'''''))), s(0)) -> ACKIN(s(s(s(m'''''))), 0)
U21(ackout(s(0)), s(s(m'''))) -> ACKIN(s(s(m''')), s(0))
ACKIN(s(s(m')), s(0)) -> U21(u11(u21(u11(ackin(m', s(0))), m')), s(m'))
U21(ackout(s(0)), s(s(s(m''''''')))) -> ACKIN(s(s(s(m'''''''))), s(0))

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

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(m''''))) -> ACKIN(s(s(m'''')), 0)
four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

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(n''''))), s(m'''')) -> ACKIN(s(m''''), s(s(n'''')))
13 new Dependency Pairs are created:

U21(ackout(s(s(0))), s(m'''''')) -> ACKIN(s(m''''''), s(s(0)))
U21(ackout(s(s(s(n''')))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(n'''))))
U21(ackout(s(s(0))), s(s(m''''''))) -> ACKIN(s(s(m'''''')), s(s(0)))
U21(ackout(s(s(s(0)))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(0))))
U21(ackout(s(s(s(s(n''''''))))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(s(n'''''')))))
U21(ackout(s(s(s(0)))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(0))))
U21(ackout(s(s(s(s(n''''''))))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(s(n'''''')))))
U21(ackout(s(s(s(n'''''')))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(n''''''))))
U21(ackout(s(s(s(0)))), s(s(m'''''''))) -> ACKIN(s(s(m''''''')), s(s(s(0))))
U21(ackout(s(s(s(s(0))))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(s(s(n''''''')))))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(s(s(n'''''''))))))
U21(ackout(s(s(0))), s(s(m''''''''))) -> ACKIN(s(s(m'''''''')), s(s(0)))
U21(ackout(s(s(0))), s(s(s(m''''''''')))) -> ACKIN(s(s(s(m'''''''''))), s(s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 14`
`                 ↳Polynomial Ordering`

Dependency Pairs:

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

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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(ack_in(x1, x2)) =  0 POL(0) =  0 POL(u11(x1)) =  0 POL(u22(x1)) =  0 POL(ACK_IN(x1, x2)) =  x1 POL(U21(x1, x2)) =  x2 POL(ack_out(x1)) =  0 POL(u21(x1, x2)) =  0 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 15`
`                 ↳Dependency Graph`

Dependency Pairs:

U21(ackout(s(s(0))), s(s(s(m''''''''')))) -> ACKIN(s(s(s(m'''''''''))), s(s(0)))
U21(ackout(s(s(0))), s(s(m''''''''))) -> ACKIN(s(s(m'''''''')), s(s(0)))
U21(ackout(s(s(s(s(s(n''''''')))))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(s(s(n'''''''))))))
U21(ackout(s(s(s(s(0))))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(0)))), s(s(m'''''''))) -> ACKIN(s(s(m''''''')), s(s(s(0))))
U21(ackout(s(s(s(n'''''')))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(n''''''))))
U21(ackout(s(s(s(s(n''''''))))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(s(n'''''')))))
U21(ackout(s(s(s(0)))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(0))))
U21(ackout(s(s(s(s(n''''''))))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(s(n'''''')))))
U21(ackout(s(s(s(0)))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(m''''''))) -> ACKIN(s(s(m'''''')), s(s(0)))
U21(ackout(s(s(s(n''')))), s(m'''''')) -> ACKIN(s(m''''''), s(s(s(n'''))))
U21(ackout(s(s(0))), s(m'''''')) -> ACKIN(s(m''''''), s(s(0)))
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(m'''''''')))) -> ACKIN(s(s(s(m''''''''))), 0)
U21(ackout(0), s(s(s(m'''''')))) -> ACKIN(s(s(s(m''''''))), 0)
U21(ackout(s(0)), s(s(s(m''''''')))) -> ACKIN(s(s(s(m'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(m''''''))) -> ACKIN(s(s(m'''''')), s(0))
U21(ackout(s(s(s(s(n'''''))))), s(m''')) -> ACKIN(s(m'''), s(s(s(s(n''''')))))
ACKIN(s(m''''), s(s(s(s(s(n''''')))))) -> ACKIN(s(m''''), s(s(s(s(n''''')))))
ACKIN(s(m''''), s(s(s(s(0))))) -> ACKIN(s(m''''), s(s(s(0))))
ACKIN(s(s(m''''')), s(s(s(0)))) -> ACKIN(s(s(m''''')), s(s(0)))
ACKIN(s(m'''), s(s(s(s(n''''))))) -> ACKIN(s(m'''), s(s(s(n''''))))
ACKIN(s(m'), s(s(s(s(n'''))))) -> ACKIN(s(m'), s(s(s(n'''))))
ACKIN(s(m''''), s(s(s(n'''')))) -> ACKIN(s(m''''), s(s(n'''')))
ACKIN(s(m'''), s(s(s(0)))) -> ACKIN(s(m'''), s(s(0)))
ACKIN(s(m'), s(s(s(0)))) -> ACKIN(s(m'), s(s(0)))
U21(ackout(s(s(s(0)))), s(m''')) -> ACKIN(s(m'''), s(s(s(0))))
ACKIN(s(s(s(m'''''''))), s(s(0))) -> ACKIN(s(s(s(m'''''''))), s(0))
ACKIN(s(s(m'''''')), s(s(0))) -> ACKIN(s(s(m'''''')), s(0))
ACKIN(s(s(m'''')), s(s(0))) -> ACKIN(s(s(m'''')), s(0))
ACKIN(s(s(m''')), s(s(0))) -> ACKIN(s(s(m''')), s(0))
U21(ackout(s(s(0))), s(s(m'''''))) -> ACKIN(s(s(m''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(0), s(s(s(m''''')))) -> ACKIN(s(s(s(m'''''))), 0)
U21(ackout(s(s(s(n''')))), s(m''''')) -> ACKIN(s(m'''''), s(s(s(n'''))))
U21(ackout(s(s(0))), s(m''''')) -> ACKIN(s(m'''''), s(s(0)))
ACKIN(s(s(s(s(m''''''''')))), s(0)) -> ACKIN(s(s(s(s(m''''''''')))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(m''''''''))), s(0)) -> ACKIN(s(s(s(m''''''''))), 0)
ACKIN(s(s(s(m''''''))), s(0)) -> ACKIN(s(s(s(m''''''))), 0)
ACKIN(s(s(s(m'''''))), s(0)) -> ACKIN(s(s(s(m'''''))), 0)
U21(ackout(s(0)), s(s(m'''))) -> ACKIN(s(s(m''')), s(0))
U21(ackout(0), s(s(s(s(m'''''''''))))) -> ACKIN(s(s(s(s(m''''''''')))), 0)

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 16`
`                 ↳Polynomial Ordering`

Dependency Pairs:

ACKIN(s(m''''), s(s(s(s(s(n''''')))))) -> ACKIN(s(m''''), s(s(s(s(n''''')))))
ACKIN(s(m''''), s(s(s(s(0))))) -> ACKIN(s(m''''), s(s(s(0))))
ACKIN(s(m''''), s(s(s(n'''')))) -> ACKIN(s(m''''), s(s(n'''')))
ACKIN(s(m'''), s(s(s(s(n''''))))) -> ACKIN(s(m'''), s(s(s(n''''))))
ACKIN(s(m'), s(s(s(s(n'''))))) -> ACKIN(s(m'), s(s(s(n'''))))

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACKIN(s(m''''), s(s(s(s(s(n''''')))))) -> ACKIN(s(m''''), s(s(s(s(n''''')))))
ACKIN(s(m''''), s(s(s(s(0))))) -> ACKIN(s(m''''), s(s(s(0))))
ACKIN(s(m''''), s(s(s(n'''')))) -> ACKIN(s(m''''), s(s(n'''')))
ACKIN(s(m'''), s(s(s(s(n''''))))) -> ACKIN(s(m'''), s(s(s(n''''))))
ACKIN(s(m'), s(s(s(s(n'''))))) -> ACKIN(s(m'), s(s(s(n'''))))

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 17`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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