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:

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


Additionally, the following usable rules for innermost can be oriented:

u21(ackout(n), m) -> u22(ackin(m, n))
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)
u22(ackout(n)) -> ackout(n)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ack_in(x1, x2))=  0  
  POL(0)=  1  
  POL(u11(x1))=  x1  
  POL(u22(x1))=  x1  
  POL(ACK_IN(x1, x2))=  0  
  POL(U21(x1, x2))=  x1  
  POL(ack_out(x1))=  x1  
  POL(u21(x1, x2))=  0  
  POL(s(x1))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 15
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(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)))
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))


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))


Additionally, the following usable rules for innermost can be oriented:

u21(ackout(n), m) -> u22(ackin(m, n))
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)
u22(ackout(n)) -> ackout(n)


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 16
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(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(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))


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 17
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 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 18
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:
0:17 minutes