Term Rewriting System R:
[X, XS, X1, X2]
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ATAIL(cons(X, XS)) -> MARK(XS)
MARK(zeros) -> AZEROS
MARK(tail(X)) -> ATAIL(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))
ATAIL(cons(X, XS)) -> MARK(XS)


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(tail(X)) -> ATAIL(mark(X))
four new Dependency Pairs are created:

MARK(tail(zeros)) -> ATAIL(azeros)
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(0)) -> ATAIL(0)

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> ATAIL(azeros)
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(tail(zeros)) -> ATAIL(azeros)
two new Dependency Pairs are created:

MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
MARK(tail(zeros)) -> ATAIL(zeros)

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Narrowing Transformation


Dependency Pairs:

MARK(tail(zeros)) -> ATAIL(cons(0, zeros))
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
five new Dependency Pairs are created:

MARK(tail(tail(X'''))) -> ATAIL(tail(mark(X''')))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(0))) -> ATAIL(atail(0))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 4
Forward Instantiation Transformation


Dependency Pairs:

MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> ATAIL(cons(0, zeros))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

ATAIL(cons(X, XS)) -> MARK(XS)
eight new Dependency Pairs are created:

ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(zeros))) -> MARK(tail(zeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))

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:

ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(zeros))) -> MARK(tail(zeros))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(tail(0))) -> ATAIL(atail(0))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(tail(X)) -> MARK(X)
seven new Dependency Pairs are created:

MARK(tail(tail(X''))) -> MARK(tail(X''))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(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:

MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(X''))) -> MARK(tail(X''))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(cons(X1, X2)) -> MARK(X1)
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(tail(0))) -> ATAIL(atail(0))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(cons(X1, X2)) -> MARK(X1)
13 new Dependency Pairs are created:

MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(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:

MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(tail(tail(X''))) -> MARK(tail(X''))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
12 new Dependency Pairs are created:

ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(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:

MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(X''))) -> MARK(tail(X''))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
13 new Dependency Pairs are created:

ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))

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:

ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(X''))) -> MARK(tail(X''))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(tail(tail(X''))) -> MARK(tail(X''))
12 new Dependency Pairs are created:

MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
MARK(tail(tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(0))))) -> MARK(tail(tail(tail(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:

ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
MARK(tail(tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
13 new Dependency Pairs are created:

MARK(tail(cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(tail(cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(tail(cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(tail(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(tail(cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(tail(cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(tail(cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(tail(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(tail(cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(tail(cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))

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:

ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(tail(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(tail(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(tail(cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
MARK(tail(tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(tail(cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(tail(tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(tail(cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(tail(cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(tail(cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
MARK(tail(cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(cons(cons(X1'', X2''), X2)) -> MARK(cons(X1'', X2''))
13 new Dependency Pairs are created:

MARK(cons(cons(cons(X1'''', X2'''''), X2''''), X2)) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(cons(cons(tail(cons(X1''''', X2''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(tail(tail(zeros)), X2'''), X2)) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(cons(cons(tail(tail(tail(X'''''))), X2'''), X2)) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''', X2'''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(cons(cons(tail(tail(0)), X2'''), X2)) -> MARK(cons(tail(tail(0)), X2'''))
MARK(cons(cons(tail(tail(X'''''')), X2'''), X2)) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(cons(tail(cons(X1'''''', X2'''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(cons(cons(tail(tail(tail(zeros))), X2'''), X2)) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(cons(cons(tail(tail(tail(tail(X''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(0))), X2'''), X2)) -> MARK(cons(tail(tail(tail(0))), X2'''))

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:

ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(tail(tail(tail(0))), X2'''), X2)) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(cons(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(tail(X''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(zeros))), X2'''), X2)) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(cons(cons(tail(cons(X1'''''', X2'''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(cons(cons(tail(tail(X'''''')), X2'''), X2)) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(cons(tail(tail(0)), X2'''), X2)) -> MARK(cons(tail(tail(0)), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''', X2'''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(cons(cons(tail(tail(tail(X'''''))), X2'''), X2)) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(cons(tail(tail(zeros)), X2'''), X2)) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(tail(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(tail(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(tail(cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
MARK(tail(tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(tail(cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(tail(tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(tail(cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(tail(cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(tail(cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(tail(cons(X1''''', X2''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(cons(X1'''', X2'''''), X2''''), X2)) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




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

MARK(cons(tail(tail(X'''')), X2)) -> MARK(tail(tail(X'''')))
16 new Dependency Pairs are created:

MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(cons(tail(tail(tail(tail(X'''''')))), X2)) -> MARK(tail(tail(tail(tail(X'''''')))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(X''''''))), X2)) -> MARK(tail(tail(tail(X''''''))))
MARK(cons(tail(tail(cons(X1'''''', X2''''''))), X2)) -> MARK(tail(tail(cons(X1'''''', X2''''''))))
MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''''', X2''''''')))))
MARK(cons(tail(tail(tail(tail(zeros)))), X2)) -> MARK(tail(tail(tail(tail(zeros)))))
MARK(cons(tail(tail(tail(tail(tail(X'''''''))))), X2)) -> MARK(tail(tail(tail(tail(tail(X'''''''))))))
MARK(cons(tail(tail(tail(tail(cons(X1''''''', X2'''''''))))), X2)) -> MARK(tail(tail(tail(tail(cons(X1''''''', X2'''''''))))))
MARK(cons(tail(tail(tail(tail(0)))), X2)) -> MARK(tail(tail(tail(tail(0)))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 13
Polynomial Ordering


Dependency Pairs:

ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(tail(tail(tail(0))), X2'''), X2)) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(cons(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(tail(X''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(zeros))), X2'''), X2)) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(cons(cons(tail(cons(X1'''''', X2'''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(cons(cons(tail(tail(X'''''')), X2'''), X2)) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(cons(tail(tail(0)), X2'''), X2)) -> MARK(cons(tail(tail(0)), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''', X2'''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(cons(cons(tail(tail(tail(X'''''))), X2'''), X2)) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(cons(tail(tail(zeros)), X2'''), X2)) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(tail(cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''''', X2''''''')))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(cons(tail(tail(tail(tail(0)))), X2)) -> MARK(tail(tail(tail(tail(0)))))
MARK(cons(tail(tail(tail(tail(cons(X1''''''', X2'''''''))))), X2)) -> MARK(tail(tail(tail(tail(cons(X1''''''', X2'''''''))))))
MARK(cons(tail(tail(tail(tail(tail(X'''''''))))), X2)) -> MARK(tail(tail(tail(tail(tail(X'''''''))))))
MARK(cons(tail(tail(tail(tail(zeros)))), X2)) -> MARK(tail(tail(tail(tail(zeros)))))
MARK(cons(tail(tail(tail(tail(X'''''')))), X2)) -> MARK(tail(tail(tail(tail(X'''''')))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(cons(tail(tail(tail(X''''''))), X2)) -> MARK(tail(tail(tail(X''''''))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
MARK(tail(tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
MARK(tail(cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(cons(tail(tail(cons(X1'''''', X2''''''))), X2)) -> MARK(tail(tail(cons(X1'''''', X2''''''))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(tail(cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(tail(tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(tail(cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(tail(cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(tail(cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(tail(cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(tail(cons(X1''''', X2''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(cons(X1'''', X2'''''), X2''''), X2)) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MARK(tail(cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(tail(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(tail(cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(tail(tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
MARK(tail(tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(tail(tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(tail(tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
MARK(tail(tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
MARK(tail(cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(tail(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(tail(cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(tail(tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
MARK(tail(cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(tail(cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(tail(tail(0))) -> ATAIL(atail(0))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(tail(cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
MARK(tail(cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(tail(cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
MARK(tail(tail(cons(X1', X2')))) -> ATAIL(atail(cons(mark(X1'), X2')))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> ATAIL(atail(azeros))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
MARK(tail(tail(tail(X')))) -> ATAIL(atail(atail(mark(X'))))


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

atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
azeros -> cons(0, zeros)
azeros -> zeros


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(MARK(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(a__zeros)=  0  
  POL(tail(x1))=  1 + x1  
  POL(a__tail(x1))=  1 + x1  
  POL(mark(x1))=  x1  
  POL(zeros)=  0  
  POL(A__TAIL(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 14
Dependency Graph


Dependency Pairs:

ATAIL(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
ATAIL(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
ATAIL(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
ATAIL(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
ATAIL(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(tail(tail(tail(0))), X2'''), X2)) -> MARK(cons(tail(tail(tail(0))), X2'''))
MARK(cons(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(tail(X''''''')))), X2'''), X2)) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
MARK(cons(cons(tail(tail(tail(zeros))), X2'''), X2)) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
MARK(cons(cons(tail(cons(X1'''''', X2'''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
MARK(cons(cons(tail(tail(X'''''')), X2'''), X2)) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(cons(tail(tail(0)), X2'''), X2)) -> MARK(cons(tail(tail(0)), X2'''))
MARK(cons(cons(tail(tail(cons(X1''''', X2'''''))), X2'''), X2)) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
MARK(cons(cons(tail(tail(tail(X'''''))), X2'''), X2)) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
MARK(cons(cons(tail(tail(zeros)), X2'''), X2)) -> MARK(cons(tail(tail(zeros)), X2'''))
ATAIL(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
ATAIL(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
ATAIL(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
ATAIL(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
ATAIL(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
ATAIL(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))
MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''''', X2''''''')))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
MARK(cons(tail(tail(tail(tail(0)))), X2)) -> MARK(tail(tail(tail(tail(0)))))
MARK(cons(tail(tail(tail(tail(cons(X1''''''', X2'''''''))))), X2)) -> MARK(tail(tail(tail(tail(cons(X1''''''', X2'''''''))))))
MARK(cons(tail(tail(tail(tail(tail(X'''''''))))), X2)) -> MARK(tail(tail(tail(tail(tail(X'''''''))))))
MARK(cons(tail(tail(tail(tail(zeros)))), X2)) -> MARK(tail(tail(tail(tail(zeros)))))
MARK(cons(tail(tail(tail(tail(X'''''')))), X2)) -> MARK(tail(tail(tail(tail(X'''''')))))
MARK(cons(tail(tail(tail(tail(X''''')))), X2)) -> MARK(tail(tail(tail(tail(X''''')))))
MARK(cons(tail(tail(tail(X''''''))), X2)) -> MARK(tail(tail(tail(X''''''))))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
MARK(cons(tail(tail(cons(X1'''''', X2''''''))), X2)) -> MARK(tail(tail(cons(X1'''''', X2''''''))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2)) -> MARK(tail(tail(cons(X1''''', X2'''''))))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(cons(X1''', X2'''))), X2)) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
ATAIL(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(cons(tail(tail(tail(zeros))), X2)) -> MARK(tail(tail(tail(zeros))))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(cons(tail(cons(X1'''', X2'''')), X2)) -> MARK(tail(cons(X1'''', X2'''')))
MARK(cons(tail(cons(X1''', X2''')), X2)) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(cons(tail(cons(X1''''', X2''''')), X2'''), X2)) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
MARK(cons(cons(cons(X1'''', X2'''''), X2''''), X2)) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
ATAIL(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
ATAIL(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
ATAIL(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


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 15
Polynomial Ordering


Dependency Pair:

MARK(cons(cons(cons(X1'''', X2'''''), X2''''), X2)) -> MARK(cons(cons(X1'''', X2'''''), X2''''))


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




The following dependency pair can be strictly oriented:

MARK(cons(cons(cons(X1'''', X2'''''), X2''''), X2)) -> MARK(cons(cons(X1'''', X2'''''), X2''''))


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MARK(x1))=  1 + x1  
  POL(cons(x1, x2))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 16
Dependency Graph


Dependency Pair:


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:24 minutes