Termination Proof using AProVE

Term Rewriting System R:
[X, XS, X1, X2]
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_TAIL}(cons(X, XS)) -> MARK(XS)
MARK(zeros) -> A_{_ZEROS}
MARK(tail(X)) -> A_{_TAIL}(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> A_{_TAIL}(mark(X))
A_{_TAIL}(cons(X, XS)) -> MARK(XS)

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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)) -> A_{_TAIL}(mark(X))

four new Dependency Pairs are created:

MARK(tail(zeros)) -> A_{_TAIL}(a_{_zeros})
MARK(tail(tail(X''))) -> A_{_TAIL}(a_{_tail}(mark(X'')))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
MARK(tail(0)) -> A_{_TAIL}(0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
MARK(tail(tail(X''))) -> A_{_TAIL}(a_{_tail}(mark(X'')))
A_{_TAIL}(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> A_{_TAIL}(a_{_zeros})
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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)) -> A_{_TAIL}(a_{_zeros})

two new Dependency Pairs are created:

MARK(tail(zeros)) -> A_{_TAIL}(cons(0, zeros))
MARK(tail(zeros)) -> A_{_TAIL}(zeros)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(tail(zeros)) -> A_{_TAIL}(cons(0, zeros))
MARK(tail(tail(X''))) -> A_{_TAIL}(a_{_tail}(mark(X'')))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
A_{_TAIL}(cons(X, XS)) -> MARK(XS)
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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''))) -> A_{_TAIL}(a_{_tail}(mark(X'')))

five new Dependency Pairs are created:

MARK(tail(tail(X'''))) -> A_{_TAIL}(tail(mark(X''')))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(cons(mark(X1'), X2')))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(0))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(cons(mark(X1'), X2')))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
A_{_TAIL}(cons(X, XS)) -> MARK(XS)
MARK(tail(zeros)) -> A_{_TAIL}(cons(0, zeros))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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

A_{_TAIL}(cons(X, XS)) -> MARK(XS)

eight new Dependency Pairs are created:

A_{_TAIL}(cons(X, tail(X''))) -> MARK(tail(X''))
A_{_TAIL}(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(zeros))) -> MARK(tail(zeros))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(cons(mark(X1'), X2')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
A_{_TAIL}(cons(X, tail(zeros))) -> MARK(tail(zeros))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
A_{_TAIL}(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(0))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:

     ↳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''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(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)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(tail(tail(X''))) -> MARK(tail(X''))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
MARK(cons(X1, X2)) -> MARK(X1)
A_{_TAIL}(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
A_{_TAIL}(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(0))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:

     ↳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'''))))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(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)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(tail(tail(X''))) -> MARK(tail(X''))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
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''))
A_{_TAIL}(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
A_{_TAIL}(cons(X, tail(X''))) -> MARK(tail(X''))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(0))
MARK(tail(tail(tail(0)))) -> MARK(tail(tail(0)))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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

A_{_TAIL}(cons(X, tail(X''))) -> MARK(tail(X''))

12 new Dependency Pairs are created:

A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
A_{_TAIL}(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
A_{_TAIL}(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
A_{_TAIL}(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
A_{_TAIL}(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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'''))))
A_{_TAIL}(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
A_{_TAIL}(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
A_{_TAIL}(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
A_{_TAIL}(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(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)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(cons(mark(X1'), X2')))
MARK(tail(tail(X''))) -> MARK(tail(X''))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
MARK(cons(tail(tail(zeros)), X2)) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(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''))
A_{_TAIL}(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(cons(tail(tail(tail(0))), X2)) -> MARK(tail(tail(tail(0))))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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

A_{_TAIL}(cons(X, cons(X1'', X2''))) -> MARK(cons(X1'', X2''))

13 new Dependency Pairs are created:

A_{_TAIL}(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
A_{_TAIL}(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

Dependency Pairs:

A_{_TAIL}(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
MARK(cons(tail(tail(0)), X2)) -> MARK(tail(tail(0)))
A_{_TAIL}(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'''))))
A_{_TAIL}(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))))
A_{_TAIL}(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'''')))
A_{_TAIL}(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'''')))
A_{_TAIL}(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
A_{_TAIL}(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
A_{_TAIL}(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
A_{_TAIL}(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
A_{_TAIL}(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
A_{_TAIL}(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(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''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(cons(mark(X1'), X2')))
MARK(tail(tail(X''))) -> MARK(tail(X''))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(cons(tail(tail(tail(cons(X1''''', X2''''')))), X2)) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

Dependency Pairs:

A_{_TAIL}(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
A_{_TAIL}(cons(X, cons(tail(cons(X1''''', X2''''')), X2'''))) -> MARK(cons(tail(cons(X1''''', X2''''')), X2'''))
A_{_TAIL}(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
A_{_TAIL}(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
A_{_TAIL}(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
A_{_TAIL}(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
A_{_TAIL}(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
A_{_TAIL}(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'''))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(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''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(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)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
A_{_TAIL}(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

Dependency Pairs:

A_{_TAIL}(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
A_{_TAIL}(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'''))
A_{_TAIL}(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
A_{_TAIL}(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
A_{_TAIL}(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
A_{_TAIL}(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
A_{_TAIL}(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
A_{_TAIL}(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'''))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_tail}(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''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(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)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
A_{_TAIL}(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Polynomial Ordering

Dependency Pairs:

A_{_TAIL}(cons(X, cons(tail(tail(tail(0))), X2'''))) -> MARK(cons(tail(tail(tail(0))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(tail(X''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(tail(X''''''')))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(zeros))), X2'''))) -> MARK(cons(tail(tail(tail(zeros))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''''', X2'''''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(cons(X1'''''', X2'''''')), X2'''))) -> MARK(cons(tail(cons(X1'''''', X2'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(X'''''')), X2'''))) -> MARK(cons(tail(tail(X'''''')), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(0)), X2'''))) -> MARK(cons(tail(tail(0)), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(cons(X1''''', X2'''''))), X2'''))) -> MARK(cons(tail(tail(cons(X1''''', X2'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(tail(X'''''))), X2'''))) -> MARK(cons(tail(tail(tail(X'''''))), X2'''))
A_{_TAIL}(cons(X, cons(tail(tail(zeros)), X2'''))) -> MARK(cons(tail(tail(zeros)), X2'''))
A_{_TAIL}(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'''))
A_{_TAIL}(cons(X, cons(cons(X1'''', X2'''''), X2''''))) -> MARK(cons(cons(X1'''', X2'''''), X2''''))
A_{_TAIL}(cons(X, tail(tail(tail(0))))) -> MARK(tail(tail(tail(0))))
A_{_TAIL}(cons(X, tail(tail(tail(cons(X1''''', X2''''')))))) -> MARK(tail(tail(tail(cons(X1''''', X2''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(tail(X''''')))))) -> MARK(tail(tail(tail(tail(X''''')))))
A_{_TAIL}(cons(X, tail(tail(tail(zeros))))) -> MARK(tail(tail(tail(zeros))))
A_{_TAIL}(cons(X, tail(tail(cons(X1''''', X2'''''))))) -> MARK(tail(tail(cons(X1''''', X2'''''))))
A_{_TAIL}(cons(X, tail(cons(X1'''', X2'''')))) -> MARK(tail(cons(X1'''', X2'''')))
A_{_TAIL}(cons(X, tail(tail(X'''')))) -> MARK(tail(tail(X'''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
A_{_TAIL}(cons(X, tail(tail(tail(X''''))))) -> MARK(tail(tail(tail(X''''))))
A_{_TAIL}(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'''))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
A_{_TAIL}(cons(X, tail(tail(0)))) -> MARK(tail(tail(0)))
MARK(tail(tail(0))) -> A_{_TAIL}(a_{_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'''))
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''')))
A_{_TAIL}(cons(X, tail(tail(cons(X1''', X2'''))))) -> MARK(tail(tail(cons(X1''', X2'''))))
MARK(tail(tail(cons(X1', X2')))) -> A_{_TAIL}(a_{_tail}(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)))
A_{_TAIL}(cons(X, tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
A_{_TAIL}(cons(X, tail(tail(zeros)))) -> MARK(tail(tail(zeros)))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
A_{_TAIL}(cons(X, tail(cons(X1''', X2''')))) -> MARK(tail(cons(X1''', X2''')))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))
MARK(cons(tail(tail(tail(X'''))), X2)) -> MARK(tail(tail(tail(X'''))))
A_{_TAIL}(cons(X, cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))) -> MARK(cons(tail(tail(tail(cons(X1''''''', X2''''''')))), X2'''))

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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))) -> A_{_TAIL}(a_{_tail}(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')))) -> A_{_TAIL}(a_{_tail}(cons(mark(X1'), X2')))
MARK(tail(tail(tail(tail(X'''))))) -> MARK(tail(tail(tail(X'''))))
MARK(tail(tail(zeros))) -> A_{_TAIL}(a_{_tail}(a_{_zeros}))
MARK(tail(cons(X1', X2'))) -> A_{_TAIL}(cons(mark(X1'), X2'))
MARK(tail(tail(tail(X')))) -> A_{_TAIL}(a_{_tail}(a_{_tail}(mark(X'))))

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

a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> 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.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Dependency Graph

Dependency Pairs:

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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.

     ↳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:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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 w.r.t. to the implicit AFS 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.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Dependency Graph

Dependency Pair:

Rules:

a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_tail}(cons(X, XS)) -> mark(XS)
a_{_tail}(X) -> tail(X)
mark(zeros) -> a_{_zeros}
mark(tail(X)) -> a_{_tail}(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:
0:25 minutes

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