Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACK_IN(s(m), 0) -> U11(ack_in(m, s(0)))
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
ACK_IN(s(m), s(n)) -> U21(ack_in(s(m), n), m)
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
U21(ack_out(n), m) -> U22(ack_in(m, n))
U21(ack_out(n), m) -> ACK_IN(m, n)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(m), s(n)) -> U21(ack_in(s(m), n), m)
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(m), s(n)) -> U21(ack_in(s(m), n), m)

two new Dependency Pairs are created:

ACK_IN(s(m''), s(0)) -> U21(u11(ack_in(m'', s(0))), m'')
ACK_IN(s(m''), s(s(n''))) -> U21(u21(ack_in(s(m''), n''), m''), m'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

ACK_IN(s(m''), s(s(n''))) -> U21(u21(ack_in(s(m''), n''), m''), m'')
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(m''), s(0)) -> U21(u11(ack_in(m'', s(0))), m'')
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(m''), s(0)) -> U21(u11(ack_in(m'', s(0))), m'')

two new Dependency Pairs are created:

ACK_IN(s(0), s(0)) -> U21(u11(ack_out(s(s(0)))), 0)
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(ack_in(s(m'), 0), m')), s(m'))

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:

ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(ack_in(s(m'), 0), m')), s(m'))
ACK_IN(s(0), s(0)) -> U21(u11(ack_out(s(s(0)))), 0)
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(m''), s(s(n''))) -> U21(u21(ack_in(s(m''), n''), m''), m'')

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(0), s(0)) -> U21(u11(ack_out(s(s(0)))), 0)

one new Dependency Pair is created:

ACK_IN(s(0), s(0)) -> U21(ack_out(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:

ACK_IN(s(0), s(0)) -> U21(ack_out(s(s(0))), 0)
ACK_IN(s(m''), s(s(n''))) -> U21(u21(ack_in(s(m''), n''), m''), m'')
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(ack_in(s(m'), 0), m')), s(m'))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(ack_in(s(m'), 0), m')), s(m'))

one new Dependency Pair is created:

ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))

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:

ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
ACK_IN(s(m''), s(s(n''))) -> U21(u21(ack_in(s(m''), n''), m''), m'')
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(0), s(0)) -> U21(ack_out(s(s(0))), 0)

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(m''), s(s(n''))) -> U21(u21(ack_in(s(m''), n''), m''), m'')

two new Dependency Pairs are created:

ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')

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:

ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
ACK_IN(s(0), s(0)) -> U21(ack_out(s(s(0))), 0)
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(m), 0) -> ACK_IN(m, s(0))

three new Dependency Pairs are created:

ACK_IN(s(s(m'')), 0) -> ACK_IN(s(m''), s(0))
ACK_IN(s(s(0)), 0) -> ACK_IN(s(0), s(0))
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), 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:

ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(0)), 0) -> ACK_IN(s(0), s(0))
ACK_IN(s(s(m'')), 0) -> ACK_IN(s(m''), s(0))
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
ACK_IN(s(0), s(0)) -> U21(ack_out(s(s(0))), 0)
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)

eight new Dependency Pairs are created:

ACK_IN(s(m''), s(s(n''))) -> ACK_IN(s(m''), s(n''))
ACK_IN(s(0), s(s(0))) -> ACK_IN(s(0), s(0))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(s(m'''')), s(0)) -> ACK_IN(s(s(m'''')), 0)
ACK_IN(s(s(0)), s(0)) -> ACK_IN(s(s(0)), 0)
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 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:

ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(0), s(s(0))) -> ACK_IN(s(0), s(0))
ACK_IN(s(m''), s(s(n''))) -> ACK_IN(s(m''), s(n''))
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
ACK_IN(s(s(0)), s(0)) -> ACK_IN(s(s(0)), 0)
ACK_IN(s(s(0)), 0) -> ACK_IN(s(0), s(0))
ACK_IN(s(s(m'''')), s(0)) -> ACK_IN(s(s(m'''')), 0)
ACK_IN(s(s(m'')), 0) -> ACK_IN(s(m''), s(0))
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
ACK_IN(s(0), s(0)) -> U21(ack_out(s(s(0))), 0)
U21(ack_out(n), m) -> ACK_IN(m, n)
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

U21(ack_out(n), m) -> ACK_IN(m, n)

15 new Dependency Pairs are created:

U21(ack_out(s(0)), s(0)) -> ACK_IN(s(0), s(0))
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
U21(ack_out(0), s(s(m''''))) -> ACK_IN(s(s(m'''')), 0)
U21(ack_out(0), s(s(0))) -> ACK_IN(s(s(0)), 0)
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(s(s(n''''))), s(m'''')) -> ACK_IN(s(m''''), s(s(n'''')))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))
U21(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), 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(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), s(0))
U21(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
U21(ack_out(s(s(n''''))), s(m'''')) -> ACK_IN(s(m''''), s(s(n'''')))
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(0), s(s(0))) -> ACK_IN(s(s(0)), 0)
U21(ack_out(0), s(s(m''''))) -> ACK_IN(s(s(m'''')), 0)
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
ACK_IN(s(s(0)), s(0)) -> ACK_IN(s(s(0)), 0)
ACK_IN(s(s(m'')), 0) -> ACK_IN(s(m''), s(0))
ACK_IN(s(s(m'''')), s(0)) -> ACK_IN(s(s(m'''')), 0)
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(m''), s(s(n''))) -> ACK_IN(s(m''), s(n''))
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

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

four new Dependency Pairs are created:

ACK_IN(s(s(s(m''''))), 0) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(s(m''''''))), 0) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(0))), 0) -> ACK_IN(s(s(0)), s(0))
ACK_IN(s(s(s(s(m''''''')))), 0) -> ACK_IN(s(s(s(m'''''''))), 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:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

ACK_IN(s(m''), s(s(n''))) -> ACK_IN(s(m''), s(n''))

nine new Dependency Pairs are created:

ACK_IN(s(s(m'''')), s(s(0))) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(m'''), s(s(s(0)))) -> ACK_IN(s(m'''), s(s(0)))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(s(m''''')), s(s(s(0)))) -> ACK_IN(s(s(m''''')), s(s(0)))
ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(s(s(s(n''''')))))
ACK_IN(s(s(m'''''')), s(s(0))) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(m'''''''))), s(s(0))) -> ACK_IN(s(s(s(m'''''''))), 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(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), s(0))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(s(s(s(n''''')))))
ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(s(m''''')), s(s(s(0)))) -> ACK_IN(s(s(m''''')), s(s(0)))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m'''), s(s(s(0)))) -> ACK_IN(s(m'''), s(s(0)))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(s(s(m'''''''))), s(s(0))) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(m'''''')), s(s(0))) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(m'''')), s(s(0))) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(s(s(n''''))), s(m'''')) -> ACK_IN(s(m''''), s(s(n'''')))
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(0), s(s(m''''))) -> ACK_IN(s(s(m'''')), 0)
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
ACK_IN(s(s(s(s(m''''''')))), 0) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(s(0))), 0) -> ACK_IN(s(s(0)), s(0))
ACK_IN(s(s(s(m''''''))), 0) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(m''''))), 0) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(m'''')), s(0)) -> ACK_IN(s(s(m'''')), 0)
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
U21(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

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

four new Dependency Pairs are created:

ACK_IN(s(s(s(m''''''))), s(0)) -> ACK_IN(s(s(s(m''''''))), 0)
ACK_IN(s(s(s(m''''''''))), s(0)) -> ACK_IN(s(s(s(m''''''''))), 0)
ACK_IN(s(s(s(0))), s(0)) -> ACK_IN(s(s(s(0))), 0)
ACK_IN(s(s(s(s(m''''''''')))), s(0)) -> ACK_IN(s(s(s(s(m''''''''')))), 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(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(s(s(s(n''''')))))
ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(s(m''''')), s(s(s(0)))) -> ACK_IN(s(s(m''''')), s(s(0)))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m'''), s(s(s(0)))) -> ACK_IN(s(m'''), s(s(0)))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(s(s(m'''''''))), s(s(0))) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(m'''''')), s(s(0))) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(m'''')), s(s(0))) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(s(s(n''''))), s(m'''')) -> ACK_IN(s(m''''), s(s(n'''')))
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(0), s(s(m''''))) -> ACK_IN(s(s(m'''')), 0)
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
ACK_IN(s(s(s(s(m''''''')))), 0) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(s(s(m''''''''')))), s(0)) -> ACK_IN(s(s(s(s(m''''''''')))), 0)
ACK_IN(s(s(s(0))), 0) -> ACK_IN(s(s(0)), s(0))
ACK_IN(s(s(s(0))), s(0)) -> ACK_IN(s(s(s(0))), 0)
ACK_IN(s(s(s(m''''''))), 0) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(m''''''''))), s(0)) -> ACK_IN(s(s(s(m''''''''))), 0)
ACK_IN(s(s(s(m''''))), 0) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(s(m''''''))), s(0)) -> ACK_IN(s(s(s(m''''''))), 0)
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
U21(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), s(0))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

U21(ack_out(0), s(s(m''''))) -> ACK_IN(s(s(m'''')), 0)

four new Dependency Pairs are created:

U21(ack_out(0), s(s(s(m'''''')))) -> ACK_IN(s(s(s(m''''''))), 0)
U21(ack_out(0), s(s(s(m'''''''')))) -> ACK_IN(s(s(s(m''''''''))), 0)
U21(ack_out(0), s(s(s(0)))) -> ACK_IN(s(s(s(0))), 0)
U21(ack_out(0), s(s(s(s(m'''''''''))))) -> ACK_IN(s(s(s(s(m''''''''')))), 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(ack_out(0), s(s(s(s(m'''''''''))))) -> ACK_IN(s(s(s(s(m''''''''')))), 0)
U21(ack_out(0), s(s(s(0)))) -> ACK_IN(s(s(s(0))), 0)
U21(ack_out(0), s(s(s(m'''''''')))) -> ACK_IN(s(s(s(m''''''''))), 0)
U21(ack_out(0), s(s(s(m'''''')))) -> ACK_IN(s(s(s(m''''''))), 0)
U21(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), s(0))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(s(s(s(n''''')))))
ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(s(m''''')), s(s(s(0)))) -> ACK_IN(s(s(m''''')), s(s(0)))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m'''), s(s(s(0)))) -> ACK_IN(s(m'''), s(s(0)))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(s(s(m'''''''))), s(s(0))) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(m'''''')), s(s(0))) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(m'''')), s(s(0))) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(s(s(n''''))), s(m'''')) -> ACK_IN(s(m''''), s(s(n'''')))
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
ACK_IN(s(s(s(s(m''''''')))), 0) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(s(s(m''''''''')))), s(0)) -> ACK_IN(s(s(s(s(m''''''''')))), 0)
ACK_IN(s(s(s(0))), 0) -> ACK_IN(s(s(0)), s(0))
ACK_IN(s(s(s(0))), s(0)) -> ACK_IN(s(s(s(0))), 0)
ACK_IN(s(s(s(m''''''))), 0) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(m''''''''))), s(0)) -> ACK_IN(s(s(s(m''''''''))), 0)
ACK_IN(s(s(s(m''''))), 0) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(s(m''''''))), s(0)) -> ACK_IN(s(s(s(m''''''))), 0)
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
U21(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

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

U21(ack_out(s(s(n''''))), s(m'''')) -> ACK_IN(s(m''''), s(s(n'''')))

13 new Dependency Pairs are created:

U21(ack_out(s(s(0))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(0)))
U21(ack_out(s(s(s(n''')))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(n'''))))
U21(ack_out(s(s(0))), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(s(0)))
U21(ack_out(s(s(s(0)))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(0))))
U21(ack_out(s(s(s(s(n''''''))))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(s(n'''''')))))
U21(ack_out(s(s(s(0)))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(0))))
U21(ack_out(s(s(s(s(n''''''))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(n'''''')))))
U21(ack_out(s(s(s(n'''''')))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(n''''''))))
U21(ack_out(s(s(s(0)))), s(s(m'''''''))) -> ACK_IN(s(s(m''''''')), s(s(s(0))))
U21(ack_out(s(s(s(s(0))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(0)))))
U21(ack_out(s(s(s(s(s(n''''''')))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(s(n'''''''))))))
U21(ack_out(s(s(0))), s(s(m''''''''))) -> ACK_IN(s(s(m'''''''')), s(s(0)))
U21(ack_out(s(s(0))), s(s(s(m''''''''')))) -> ACK_IN(s(s(s(m'''''''''))), 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(ack_out(s(s(0))), s(s(s(m''''''''')))) -> ACK_IN(s(s(s(m'''''''''))), s(s(0)))
U21(ack_out(s(s(0))), s(s(m''''''''))) -> ACK_IN(s(s(m'''''''')), s(s(0)))
U21(ack_out(s(s(s(s(s(n''''''')))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(s(n'''''''))))))
U21(ack_out(s(s(s(s(0))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(0)))))
U21(ack_out(s(s(s(0)))), s(s(m'''''''))) -> ACK_IN(s(s(m''''''')), s(s(s(0))))
U21(ack_out(s(s(s(n'''''')))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(n''''''))))
U21(ack_out(s(s(s(s(n''''''))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(n'''''')))))
U21(ack_out(s(s(s(0)))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(0))))
U21(ack_out(s(s(s(s(n''''''))))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(s(n'''''')))))
U21(ack_out(s(s(s(0)))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(s(0)))
U21(ack_out(s(s(s(n''')))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(n'''))))
U21(ack_out(s(s(0))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(0)))
U21(ack_out(0), s(s(s(0)))) -> ACK_IN(s(s(s(0))), 0)
U21(ack_out(0), s(s(s(m'''''''')))) -> ACK_IN(s(s(s(m''''''''))), 0)
U21(ack_out(0), s(s(s(m'''''')))) -> ACK_IN(s(s(s(m''''''))), 0)
U21(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), s(0))
U21(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(s(s(s(n''''')))))
ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(s(m''''')), s(s(s(0)))) -> ACK_IN(s(s(m''''')), s(s(0)))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(m'''), s(s(s(0)))) -> ACK_IN(s(m'''), s(s(0)))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
ACK_IN(s(s(s(m'''''''))), s(s(0))) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(m'''''')), s(s(0))) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(m'''')), s(s(0))) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
ACK_IN(s(s(s(s(m''''''''')))), s(0)) -> ACK_IN(s(s(s(s(m''''''''')))), 0)
ACK_IN(s(s(s(0))), s(0)) -> ACK_IN(s(s(s(0))), 0)
ACK_IN(s(s(s(s(m''''''')))), 0) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(s(0))), 0) -> ACK_IN(s(s(0)), s(0))
ACK_IN(s(s(s(m''''''''))), s(0)) -> ACK_IN(s(s(s(m''''''''))), 0)
ACK_IN(s(s(s(m''''''))), 0) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(m''''''))), s(0)) -> ACK_IN(s(s(s(m''''''))), 0)
ACK_IN(s(s(s(m''''))), 0) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(0), s(s(s(s(m'''''''''))))) -> ACK_IN(s(s(s(s(m''''''''')))), 0)

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

U21(ack_out(s(s(0))), s(s(s(m''''''''')))) -> ACK_IN(s(s(s(m'''''''''))), s(s(0)))
U21(ack_out(s(s(0))), s(s(m''''''''))) -> ACK_IN(s(s(m'''''''')), s(s(0)))
U21(ack_out(s(s(s(s(s(n''''''')))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(s(n'''''''))))))
U21(ack_out(s(s(s(s(0))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(0)))))
U21(ack_out(s(s(s(0)))), s(s(m'''''''))) -> ACK_IN(s(s(m''''''')), s(s(s(0))))
U21(ack_out(s(s(s(n'''''')))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(n''''''))))
U21(ack_out(s(s(s(s(n''''''))))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(s(n'''''')))))
U21(ack_out(s(s(s(0)))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(0))))
U21(ack_out(s(s(s(s(n''''''))))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(s(n'''''')))))
U21(ack_out(s(s(s(0)))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(s(0)))
U21(ack_out(s(s(s(n''')))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(s(n'''))))
U21(ack_out(s(s(0))), s(m'''''')) -> ACK_IN(s(m''''''), s(s(0)))
U21(ack_out(0), s(s(s(0)))) -> ACK_IN(s(s(s(0))), 0)
U21(ack_out(0), s(s(s(m'''''''')))) -> ACK_IN(s(s(s(m''''''''))), 0)
U21(ack_out(0), s(s(s(m'''''')))) -> ACK_IN(s(s(s(m''''''))), 0)
U21(ack_out(s(0)), s(s(s(m''''''')))) -> ACK_IN(s(s(s(m'''''''))), s(0))
U21(ack_out(s(0)), s(s(0))) -> ACK_IN(s(s(0)), s(0))
U21(ack_out(s(0)), s(s(m''''''))) -> ACK_IN(s(s(m'''''')), s(0))
U21(ack_out(s(s(s(s(n'''''))))), s(m''')) -> ACK_IN(s(m'''), s(s(s(s(n''''')))))
U21(ack_out(s(s(s(0)))), s(m''')) -> ACK_IN(s(m'''), s(s(s(0))))
U21(ack_out(s(s(0))), s(s(m'''''))) -> ACK_IN(s(s(m''''')), s(s(0)))
U21(ack_out(s(s(0))), s(0)) -> ACK_IN(s(0), s(s(0)))
U21(ack_out(0), s(s(s(m''''')))) -> ACK_IN(s(s(s(m'''''))), 0)
U21(ack_out(s(s(s(n''')))), s(m''''')) -> ACK_IN(s(m'''''), s(s(s(n'''))))
U21(ack_out(s(s(0))), s(m''''')) -> ACK_IN(s(m'''''), s(s(0)))
ACK_IN(s(s(s(s(m''''''')))), 0) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(s(0))), 0) -> ACK_IN(s(s(0)), s(0))
ACK_IN(s(s(s(m''''''))), 0) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(s(m''''))), 0) -> ACK_IN(s(s(m'''')), s(0))
U21(ack_out(s(0)), s(s(m'''))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(s(s(m'''))), 0) -> ACK_IN(s(s(m''')), s(0))
U21(ack_out(0), s(s(s(s(m'''''''''))))) -> ACK_IN(s(s(s(s(m''''''''')))), 0)

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

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u11(ack_out(n)) -> ack_out(n)
u22(ack_out(n)) -> ack_out(n)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ack_in(x₁, x₂)) = 1

POL(0) = 0

POL(u11(x₁)) = 1

POL(u22(x₁)) = 1

POL(ACK_IN(x₁, x₂)) = x₁

POL(U21(x₁, x₂)) = x₁ + x₂

POL(ack_out(x₁)) = 1

POL(u21(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Dependency Graph

Dependency Pairs:

ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(s(s(s(n''''')))))
ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(s(m''''')), s(s(s(0)))) -> ACK_IN(s(s(m''''')), s(s(0)))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(m'''), s(s(s(0)))) -> ACK_IN(s(m'''), s(s(0)))
ACK_IN(s(m'), s(s(s(0)))) -> ACK_IN(s(m'), s(s(0)))
ACK_IN(s(s(s(m'''''''))), s(s(0))) -> ACK_IN(s(s(s(m'''''''))), s(0))
ACK_IN(s(s(m'''''')), s(s(0))) -> ACK_IN(s(s(m'''''')), s(0))
ACK_IN(s(s(m'''')), s(s(0))) -> ACK_IN(s(s(m'''')), s(0))
ACK_IN(s(s(m''')), s(s(0))) -> ACK_IN(s(s(m''')), s(0))
ACK_IN(s(m'''), s(s(s(n')))) -> U21(u21(u21(ack_in(s(m'''), n'), m'''), m'''), m''')
ACK_IN(s(m'''), s(s(0))) -> U21(u21(u11(ack_in(m''', s(0))), m'''), m''')
ACK_IN(s(s(s(s(m''''''''')))), s(0)) -> ACK_IN(s(s(s(s(m''''''''')))), 0)
ACK_IN(s(s(s(0))), s(0)) -> ACK_IN(s(s(s(0))), 0)
ACK_IN(s(s(s(m''''''''))), s(0)) -> ACK_IN(s(s(s(m''''''''))), 0)
ACK_IN(s(s(s(m''''''))), s(0)) -> ACK_IN(s(s(s(m''''''))), 0)
ACK_IN(s(s(s(m'''''))), s(0)) -> ACK_IN(s(s(s(m'''''))), 0)
ACK_IN(s(s(m')), s(0)) -> U21(u11(u21(u11(ack_in(m', s(0))), m')), s(m'))

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

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:

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

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACK_IN(s(m''''), s(s(s(s(0))))) -> ACK_IN(s(m''''), s(s(s(0))))
ACK_IN(s(m''''), s(s(s(n'''')))) -> ACK_IN(s(m''''), s(s(n'''')))
ACK_IN(s(m'''), s(s(s(s(n''''))))) -> ACK_IN(s(m'''), s(s(s(n''''))))
ACK_IN(s(m'), s(s(s(s(n'''))))) -> ACK_IN(s(m'), s(s(s(n'''))))
ACK_IN(s(m''''), s(s(s(s(s(n''''')))))) -> ACK_IN(s(m''''), s(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.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 17

                 ↳Dependency Graph

Dependency Pair:

Rules:

ack_in(0, n) -> ack_out(s(n))
ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u11(ack_out(n)) -> ack_out(n)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

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

POL(0)	= 0
POL(ACK_IN(x₁, x₂))	= 1 + x₁ + x₂
POL(s(x₁))	= 1 + x₁