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
Forward Instantiation 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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
FwdInst
             ...
               →DP Problem 3
Forward Instantiation Transformation


Dependency Pairs:

ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(zeros))) -> MARK(tail(zeros))
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(zeros)) -> ATAIL(azeros)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

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

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
MARK(tail(tail(zeros))) -> MARK(tail(zeros))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(tail(X''))) -> MARK(tail(X''))
ATAIL(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
ATAIL(cons(X, tail(zeros))) -> MARK(tail(zeros))
MARK(tail(tail(X''))) -> ATAIL(atail(mark(X'')))
MARK(cons(X1, X2)) -> MARK(X1)
ATAIL(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(zeros)) -> ATAIL(azeros)
ATAIL(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(cons(X1', X2'))) -> ATAIL(cons(mark(X1'), X2'))
ATAIL(cons(X, tail(cons(X1''', X2''')))) -> MARK(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(cons(X1, X2)) -> MARK(X1)
eight new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
FwdInst
             ...
               →DP Problem 5
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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



Innermost Termination of R could not be shown.
Duration:
0:03 minutes