Term Rewriting System R:
[y, x]
ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACK(s(x), 0) -> ACK(x, s(0))
ACK(s(x), s(y)) -> ACK(x, ack(s(x), y))
ACK(s(x), s(y)) -> ACK(s(x), y)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

ACK(s(x), s(y)) -> ACK(s(x), y)
ACK(s(x), s(y)) -> ACK(x, ack(s(x), y))
ACK(s(x), 0) -> ACK(x, s(0))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(x), s(y)) -> ACK(x, ack(s(x), y))
two new Dependency Pairs are created:

ACK(s(x''), s(0)) -> ACK(x'', ack(x'', s(0)))
ACK(s(x''), s(s(y''))) -> ACK(x'', ack(x'', ack(s(x''), y'')))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACK(s(x''), s(s(y''))) -> ACK(x'', ack(x'', ack(s(x''), y'')))
ACK(s(x''), s(0)) -> ACK(x'', ack(x'', s(0)))
ACK(s(x), 0) -> ACK(x, s(0))
ACK(s(x), s(y)) -> ACK(s(x), y)


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(x''), s(0)) -> ACK(x'', ack(x'', s(0)))
two new Dependency Pairs are created:

ACK(s(0), s(0)) -> ACK(0, s(s(0)))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(s(x'), 0)))

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:

ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(s(x'), 0)))
ACK(s(x), s(y)) -> ACK(s(x), y)
ACK(s(x), 0) -> ACK(x, s(0))
ACK(s(x''), s(s(y''))) -> ACK(x'', ack(x'', ack(s(x''), y'')))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(s(x'), 0)))
one new Dependency Pair is created:

ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 4
Narrowing Transformation


Dependency Pairs:

ACK(s(x''), s(s(y''))) -> ACK(x'', ack(x'', ack(s(x''), y'')))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(x), 0) -> ACK(x, s(0))
ACK(s(x), s(y)) -> ACK(s(x), y)


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(x''), s(s(y''))) -> ACK(x'', ack(x'', ack(s(x''), y'')))
three new Dependency Pairs are created:

ACK(s(0), s(s(y'''))) -> ACK(0, s(ack(s(0), y''')))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 5
Forward Instantiation Transformation


Dependency Pairs:

ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(x), s(y)) -> ACK(s(x), y)
ACK(s(x), 0) -> ACK(x, s(0))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(x), 0) -> ACK(x, s(0))
two new Dependency Pairs are created:

ACK(s(s(x'')), 0) -> ACK(s(x''), s(0))
ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))

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:

ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))
ACK(s(s(x'')), 0) -> ACK(s(x''), s(0))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(x), s(y)) -> ACK(s(x), y)
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(x), s(y)) -> ACK(s(x), y)
six new Dependency Pairs are created:

ACK(s(x''), s(s(y''))) -> ACK(s(x''), s(y''))
ACK(s(s(x''')), s(s(0))) -> ACK(s(s(x''')), s(0))
ACK(s(x'), s(s(s(0)))) -> ACK(s(x'), s(s(0)))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(s(x'''')), s(0)) -> ACK(s(s(x'''')), 0)
ACK(s(s(s(x'''''))), s(0)) -> ACK(s(s(s(x'''''))), 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:

ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(x'), s(s(s(0)))) -> ACK(s(x'), s(s(0)))
ACK(s(s(x''')), s(s(0))) -> ACK(s(s(x''')), s(0))
ACK(s(x''), s(s(y''))) -> ACK(s(x''), s(y''))
ACK(s(s(s(x'''''))), s(0)) -> ACK(s(s(s(x'''''))), 0)
ACK(s(s(x'''')), s(0)) -> ACK(s(s(x'''')), 0)
ACK(s(s(x'')), 0) -> ACK(s(x''), s(0))
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(s(x'')), 0) -> ACK(s(x''), s(0))
three new Dependency Pairs are created:

ACK(s(s(s(x''''))), 0) -> ACK(s(s(x'''')), s(0))
ACK(s(s(s(x''''''))), 0) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(s(s(x''''''')))), 0) -> ACK(s(s(s(x'''''''))), s(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:

ACK(s(x'), s(s(s(0)))) -> ACK(s(x'), s(s(0)))
ACK(s(s(x''')), s(s(0))) -> ACK(s(s(x''')), s(0))
ACK(s(x''), s(s(y''))) -> ACK(s(x''), s(y''))
ACK(s(s(s(s(x''''''')))), 0) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(s(x''''''))), 0) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(s(x'''''))), s(0)) -> ACK(s(s(s(x'''''))), 0)
ACK(s(s(s(x''''))), 0) -> ACK(s(s(x'''')), s(0))
ACK(s(s(x'''')), s(0)) -> ACK(s(s(x'''')), 0)
ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(x''), s(s(y''))) -> ACK(s(x''), s(y''))
nine new Dependency Pairs are created:

ACK(s(s(x'''')), s(s(0))) -> ACK(s(s(x'''')), s(0))
ACK(s(x'''), s(s(s(0)))) -> ACK(s(x'''), s(s(0)))
ACK(s(x'''), s(s(s(s(y''''))))) -> ACK(s(x'''), s(s(s(y''''))))
ACK(s(x''''), s(s(s(y'''')))) -> ACK(s(x''''), s(s(y'''')))
ACK(s(s(x''''')), s(s(s(0)))) -> ACK(s(s(x''''')), s(s(0)))
ACK(s(x''''), s(s(s(s(0))))) -> ACK(s(x''''), s(s(s(0))))
ACK(s(x''''), s(s(s(s(s(y''''')))))) -> ACK(s(x''''), s(s(s(s(y''''')))))
ACK(s(s(x'''''')), s(s(0))) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(s(x'''''''))), s(s(0))) -> ACK(s(s(s(x'''''''))), 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:

ACK(s(x''''), s(s(s(s(s(y''''')))))) -> ACK(s(x''''), s(s(s(s(y''''')))))
ACK(s(x''''), s(s(s(s(0))))) -> ACK(s(x''''), s(s(s(0))))
ACK(s(s(x''''')), s(s(s(0)))) -> ACK(s(s(x''''')), s(s(0)))
ACK(s(x''''), s(s(s(y'''')))) -> ACK(s(x''''), s(s(y'''')))
ACK(s(x'''), s(s(s(s(y''''))))) -> ACK(s(x'''), s(s(s(y''''))))
ACK(s(s(s(x'''''''))), s(s(0))) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(x'''''')), s(s(0))) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(x'''')), s(s(0))) -> ACK(s(s(x'''')), s(0))
ACK(s(x'''), s(s(s(0)))) -> ACK(s(x'''), s(s(0)))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(s(x''')), s(s(0))) -> ACK(s(s(x''')), s(0))
ACK(s(s(s(s(x''''''')))), 0) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(s(x''''''))), 0) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(s(x'''''))), s(0)) -> ACK(s(s(s(x'''''))), 0)
ACK(s(s(s(x''''))), 0) -> ACK(s(s(x'''')), s(0))
ACK(s(s(x'''')), s(0)) -> ACK(s(s(x'''')), 0)
ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(x'), s(s(s(0)))) -> ACK(s(x'), s(s(0)))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




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

ACK(s(s(x'''')), s(0)) -> ACK(s(s(x'''')), 0)
three new Dependency Pairs are created:

ACK(s(s(s(x''''''))), s(0)) -> ACK(s(s(s(x''''''))), 0)
ACK(s(s(s(x''''''''))), s(0)) -> ACK(s(s(s(x''''''''))), 0)
ACK(s(s(s(s(x''''''''')))), s(0)) -> ACK(s(s(s(s(x''''''''')))), 0)

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 10
Polynomial Ordering


Dependency Pairs:

ACK(s(x''''), s(s(s(s(0))))) -> ACK(s(x''''), s(s(s(0))))
ACK(s(s(x''''')), s(s(s(0)))) -> ACK(s(s(x''''')), s(s(0)))
ACK(s(x''''), s(s(s(y'''')))) -> ACK(s(x''''), s(s(y'''')))
ACK(s(x'''), s(s(s(s(y''''))))) -> ACK(s(x'''), s(s(s(y''''))))
ACK(s(x'''), s(s(s(0)))) -> ACK(s(x'''), s(s(0)))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(s(s(x'''''''))), s(s(0))) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(x'''''')), s(s(0))) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(x'''')), s(s(0))) -> ACK(s(s(x'''')), s(0))
ACK(s(x'), s(s(s(0)))) -> ACK(s(x'), s(s(0)))
ACK(s(s(x''')), s(s(0))) -> ACK(s(s(x''')), s(0))
ACK(s(s(s(s(x''''''''')))), s(0)) -> ACK(s(s(s(s(x''''''''')))), 0)
ACK(s(s(s(s(x''''''')))), 0) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(s(x''''''''))), s(0)) -> ACK(s(s(s(x''''''''))), 0)
ACK(s(s(s(x''''''))), 0) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(s(x''''''))), s(0)) -> ACK(s(s(s(x''''''))), 0)
ACK(s(s(s(x''''))), 0) -> ACK(s(s(x'''')), s(0))
ACK(s(s(s(x'''''))), s(0)) -> ACK(s(s(s(x'''''))), 0)
ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))
ACK(s(x''''), s(s(s(s(s(y''''')))))) -> ACK(s(x''''), s(s(s(s(y''''')))))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACK(s(s(s(s(x''''''')))), 0) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(s(x''''''))), 0) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(s(x''''))), 0) -> ACK(s(s(x'''')), s(0))
ACK(s(s(s(x'''))), 0) -> ACK(s(s(x''')), s(0))
ACK(s(x'''), s(s(0))) -> ACK(x''', ack(x''', ack(x''', s(0))))
ACK(s(s(x')), s(0)) -> ACK(s(x'), ack(x', ack(x', s(0))))
ACK(s(x'''), s(s(s(y')))) -> ACK(x''', ack(x''', ack(x''', ack(s(x'''), y'))))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACK(x1, x2))=  x1  
  POL(ack(x1, x2))=  0  
  POL(0)=  0  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 11
Dependency Graph


Dependency Pairs:

ACK(s(x''''), s(s(s(s(0))))) -> ACK(s(x''''), s(s(s(0))))
ACK(s(s(x''''')), s(s(s(0)))) -> ACK(s(s(x''''')), s(s(0)))
ACK(s(x''''), s(s(s(y'''')))) -> ACK(s(x''''), s(s(y'''')))
ACK(s(x'''), s(s(s(s(y''''))))) -> ACK(s(x'''), s(s(s(y''''))))
ACK(s(x'''), s(s(s(0)))) -> ACK(s(x'''), s(s(0)))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(s(s(x'''''''))), s(s(0))) -> ACK(s(s(s(x'''''''))), s(0))
ACK(s(s(x'''''')), s(s(0))) -> ACK(s(s(x'''''')), s(0))
ACK(s(s(x'''')), s(s(0))) -> ACK(s(s(x'''')), s(0))
ACK(s(x'), s(s(s(0)))) -> ACK(s(x'), s(s(0)))
ACK(s(s(x''')), s(s(0))) -> ACK(s(s(x''')), s(0))
ACK(s(s(s(s(x''''''''')))), s(0)) -> ACK(s(s(s(s(x''''''''')))), 0)
ACK(s(s(s(x''''''''))), s(0)) -> ACK(s(s(s(x''''''''))), 0)
ACK(s(s(s(x''''''))), s(0)) -> ACK(s(s(s(x''''''))), 0)
ACK(s(s(s(x'''''))), s(0)) -> ACK(s(s(s(x'''''))), 0)
ACK(s(x''''), s(s(s(s(s(y''''')))))) -> ACK(s(x''''), s(s(s(s(y''''')))))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


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 12
Polynomial Ordering


Dependency Pairs:

ACK(s(x''''), s(s(s(s(s(y''''')))))) -> ACK(s(x''''), s(s(s(s(y''''')))))
ACK(s(x'''), s(s(s(s(y''''))))) -> ACK(s(x'''), s(s(s(y''''))))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(x''''), s(s(s(y'''')))) -> ACK(s(x''''), s(s(y'''')))
ACK(s(x''''), s(s(s(s(0))))) -> ACK(s(x''''), s(s(s(0))))


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACK(s(x''''), s(s(s(s(s(y''''')))))) -> ACK(s(x''''), s(s(s(s(y''''')))))
ACK(s(x'''), s(s(s(s(y''''))))) -> ACK(s(x'''), s(s(s(y''''))))
ACK(s(x'), s(s(s(s(y'''))))) -> ACK(s(x'), s(s(s(y'''))))
ACK(s(x''''), s(s(s(y'''')))) -> ACK(s(x''''), s(s(y'''')))
ACK(s(x''''), s(s(s(s(0))))) -> ACK(s(x''''), s(s(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(x1, x2))=  1 + x1 + x2  
  POL(0)=  0  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 13
Dependency Graph


Dependency Pair:


Rules:


ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes