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

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

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

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Innermost Termination of R could not be shown.
Duration:
0:05 minutes