Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2]
a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_NATS} -> A_{_ADX}(a_{_zeros})
A_{_NATS} -> A_{_ZEROS}
A_{_ADX}(cons(X, Y)) -> A_{_INCR}(cons(X, adx(Y)))
A_{_HD}(cons(X, Y)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(nats) -> A_{_NATS}
MARK(adx(X)) -> A_{_ADX}(mark(X))
MARK(adx(X)) -> MARK(X)
MARK(zeros) -> A_{_ZEROS}
MARK(incr(X)) -> A_{_INCR}(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(hd(X)) -> A_{_HD}(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(tl(X)) -> A_{_TL}(mark(X))
MARK(tl(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

MARK(tl(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(X)) -> A_{_TL}(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> A_{_HD}(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(X)) -> A_{_HD}(mark(X))

nine new Dependency Pairs are created:

MARK(hd(nats)) -> A_{_HD}(a_{_nats})
MARK(hd(adx(X''))) -> A_{_HD}(a_{_adx}(mark(X'')))
MARK(hd(zeros)) -> A_{_HD}(a_{_zeros})
MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(0)) -> A_{_HD}(0)
MARK(hd(s(X''))) -> A_{_HD}(s(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))
MARK(hd(zeros)) -> A_{_HD}(a_{_zeros})
MARK(hd(adx(X''))) -> A_{_HD}(a_{_adx}(mark(X'')))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(nats)) -> A_{_HD}(a_{_nats})
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(X)) -> A_{_TL}(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
MARK(tl(X)) -> MARK(X)

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(X)) -> A_{_TL}(mark(X))

nine new Dependency Pairs are created:

MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(0)) -> A_{_TL}(0)
MARK(tl(s(X''))) -> A_{_TL}(s(X''))

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(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))
MARK(hd(zeros)) -> A_{_HD}(a_{_zeros})
MARK(hd(adx(X''))) -> A_{_HD}(a_{_adx}(mark(X'')))
MARK(hd(nats)) -> A_{_HD}(a_{_nats})
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(nats)) -> A_{_HD}(a_{_nats})

two new Dependency Pairs are created:

MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(nats)) -> A_{_HD}(nats)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))
MARK(hd(zeros)) -> A_{_HD}(a_{_zeros})
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(adx(X''))) -> A_{_HD}(a_{_adx}(mark(X'')))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(adx(X''))) -> A_{_HD}(a_{_adx}(mark(X'')))

10 new Dependency Pairs are created:

MARK(hd(adx(X'''))) -> A_{_HD}(adx(mark(X''')))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))
MARK(hd(zeros)) -> A_{_HD}(a_{_zeros})
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(zeros)) -> A_{_HD}(a_{_zeros})

two new Dependency Pairs are created:

MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(zeros)) -> A_{_HD}(zeros)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(incr(X''))) -> A_{_HD}(a_{_incr}(mark(X'')))

10 new Dependency Pairs are created:

MARK(hd(incr(X'''))) -> A_{_HD}(incr(mark(X''')))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(hd(X''))) -> A_{_HD}(a_{_hd}(mark(X'')))

10 new Dependency Pairs are created:

MARK(hd(hd(X'''))) -> A_{_HD}(hd(mark(X''')))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(hd(tl(X''))) -> A_{_HD}(a_{_tl}(mark(X'')))

10 new Dependency Pairs are created:

MARK(hd(tl(X'''))) -> A_{_HD}(tl(mark(X''')))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_nats})
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(nats)) -> A_{_TL}(a_{_nats})

two new Dependency Pairs are created:

MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(nats)) -> A_{_TL}(nats)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(adx(X''))) -> A_{_TL}(a_{_adx}(mark(X'')))

10 new Dependency Pairs are created:

MARK(tl(adx(X'''))) -> A_{_TL}(adx(mark(X''')))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(zeros)) -> A_{_TL}(a_{_zeros})

two new Dependency Pairs are created:

MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(zeros)) -> A_{_TL}(zeros)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(incr(X''))) -> A_{_TL}(a_{_incr}(mark(X'')))

10 new Dependency Pairs are created:

MARK(tl(incr(X'''))) -> A_{_TL}(incr(mark(X''')))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(hd(X''))) -> A_{_TL}(a_{_hd}(mark(X'')))

10 new Dependency Pairs are created:

MARK(tl(hd(X'''))) -> A_{_TL}(hd(mark(X''')))
MARK(tl(hd(nats))) -> A_{_TL}(a_{_hd}(a_{_nats}))
MARK(tl(hd(adx(X')))) -> A_{_TL}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(tl(hd(zeros))) -> A_{_TL}(a_{_hd}(a_{_zeros}))
MARK(tl(hd(incr(X')))) -> A_{_TL}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(tl(hd(hd(X')))) -> A_{_TL}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(tl(hd(tl(X')))) -> A_{_TL}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(tl(hd(cons(X1', X2')))) -> A_{_TL}(a_{_hd}(cons(X1', X2')))
MARK(tl(hd(0))) -> A_{_TL}(a_{_hd}(0))
MARK(tl(hd(s(X')))) -> A_{_TL}(a_{_hd}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(tl(hd(s(X')))) -> A_{_TL}(a_{_hd}(s(X')))
MARK(tl(hd(0))) -> A_{_TL}(a_{_hd}(0))
MARK(tl(hd(cons(X1', X2')))) -> A_{_TL}(a_{_hd}(cons(X1', X2')))
MARK(tl(hd(tl(X')))) -> A_{_TL}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(tl(hd(hd(X')))) -> A_{_TL}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(tl(hd(incr(X')))) -> A_{_TL}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(tl(hd(zeros))) -> A_{_TL}(a_{_hd}(a_{_zeros}))
MARK(tl(hd(adx(X')))) -> A_{_TL}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(tl(hd(nats))) -> A_{_TL}(a_{_hd}(a_{_nats}))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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

MARK(tl(tl(X''))) -> A_{_TL}(a_{_tl}(mark(X'')))

10 new Dependency Pairs are created:

MARK(tl(tl(X'''))) -> A_{_TL}(tl(mark(X''')))
MARK(tl(tl(nats))) -> A_{_TL}(a_{_tl}(a_{_nats}))
MARK(tl(tl(adx(X')))) -> A_{_TL}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(tl(tl(zeros))) -> A_{_TL}(a_{_tl}(a_{_zeros}))
MARK(tl(tl(incr(X')))) -> A_{_TL}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(tl(tl(hd(X')))) -> A_{_TL}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(tl(tl(tl(X')))) -> A_{_TL}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(tl(tl(cons(X1', X2')))) -> A_{_TL}(a_{_tl}(cons(X1', X2')))
MARK(tl(tl(0))) -> A_{_TL}(a_{_tl}(0))
MARK(tl(tl(s(X')))) -> A_{_TL}(a_{_tl}(s(X')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(tl(tl(s(X')))) -> A_{_TL}(a_{_tl}(s(X')))
MARK(tl(tl(0))) -> A_{_TL}(a_{_tl}(0))
MARK(tl(tl(cons(X1', X2')))) -> A_{_TL}(a_{_tl}(cons(X1', X2')))
MARK(tl(tl(tl(X')))) -> A_{_TL}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(tl(tl(hd(X')))) -> A_{_TL}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(tl(tl(incr(X')))) -> A_{_TL}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(tl(tl(zeros))) -> A_{_TL}(a_{_tl}(a_{_zeros}))
MARK(tl(tl(adx(X')))) -> A_{_TL}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(tl(tl(nats))) -> A_{_TL}(a_{_tl}(a_{_nats}))
MARK(tl(hd(0))) -> A_{_TL}(a_{_hd}(0))
MARK(tl(hd(cons(X1', X2')))) -> A_{_TL}(a_{_hd}(cons(X1', X2')))
MARK(tl(hd(tl(X')))) -> A_{_TL}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(tl(hd(hd(X')))) -> A_{_TL}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(tl(hd(incr(X')))) -> A_{_TL}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(tl(hd(zeros))) -> A_{_TL}(a_{_hd}(a_{_zeros}))
MARK(tl(hd(adx(X')))) -> A_{_TL}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(tl(hd(nats))) -> A_{_TL}(a_{_hd}(a_{_nats}))
MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(hd(s(X')))) -> A_{_TL}(a_{_hd}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

The following dependency pairs can be strictly oriented:

MARK(tl(nats)) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(hd(nats)) -> A_{_HD}(a_{_adx}(a_{_zeros}))

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

a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(a__nats) = 1

POL(MARK(x₁)) = x₁

POL(adx(x₁)) = x₁

POL(a__zeros) = 0

POL(incr(x₁)) = x₁

POL(A__TL(x₁)) = x₁

POL(a__hd(x₁)) = x₁

POL(a__tl(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(tl(x₁)) = x₁

POL(a__adx(x₁)) = x₁

POL(A__HD(x₁)) = x₁

POL(0) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(hd(x₁)) = x₁

POL(nats) = 1

POL(s(x₁)) = 0

POL(zeros) = 0

POL(a__incr(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(tl(tl(s(X')))) -> A_{_TL}(a_{_tl}(s(X')))
MARK(tl(tl(0))) -> A_{_TL}(a_{_tl}(0))
MARK(tl(tl(cons(X1', X2')))) -> A_{_TL}(a_{_tl}(cons(X1', X2')))
MARK(tl(tl(tl(X')))) -> A_{_TL}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(tl(tl(hd(X')))) -> A_{_TL}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(tl(tl(incr(X')))) -> A_{_TL}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(tl(tl(zeros))) -> A_{_TL}(a_{_tl}(a_{_zeros}))
MARK(tl(tl(adx(X')))) -> A_{_TL}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(tl(tl(nats))) -> A_{_TL}(a_{_tl}(a_{_nats}))
MARK(tl(hd(0))) -> A_{_TL}(a_{_hd}(0))
MARK(tl(hd(cons(X1', X2')))) -> A_{_TL}(a_{_hd}(cons(X1', X2')))
MARK(tl(hd(tl(X')))) -> A_{_TL}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(tl(hd(hd(X')))) -> A_{_TL}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(tl(hd(incr(X')))) -> A_{_TL}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(tl(hd(zeros))) -> A_{_TL}(a_{_hd}(a_{_zeros}))
MARK(tl(hd(adx(X')))) -> A_{_TL}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(tl(hd(nats))) -> A_{_TL}(a_{_hd}(a_{_nats}))
MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(adx(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(hd(s(X')))) -> A_{_TL}(a_{_hd}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

The following dependency pair can be strictly oriented:

MARK(adx(X)) -> MARK(X)

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

a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(a__nats) = 1

POL(MARK(x₁)) = x₁

POL(adx(x₁)) = 1 + x₁

POL(a__zeros) = 0

POL(incr(x₁)) = x₁

POL(A__TL(x₁)) = x₁

POL(a__hd(x₁)) = x₁

POL(a__tl(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(tl(x₁)) = x₁

POL(a__adx(x₁)) = 1 + x₁

POL(A__HD(x₁)) = x₁

POL(0) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(hd(x₁)) = x₁

POL(nats) = 1

POL(s(x₁)) = 0

POL(zeros) = 0

POL(a__incr(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 17

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(tl(tl(s(X')))) -> A_{_TL}(a_{_tl}(s(X')))
MARK(tl(tl(0))) -> A_{_TL}(a_{_tl}(0))
MARK(tl(tl(cons(X1', X2')))) -> A_{_TL}(a_{_tl}(cons(X1', X2')))
MARK(tl(tl(tl(X')))) -> A_{_TL}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(tl(tl(hd(X')))) -> A_{_TL}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(tl(tl(incr(X')))) -> A_{_TL}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(tl(tl(zeros))) -> A_{_TL}(a_{_tl}(a_{_zeros}))
MARK(tl(tl(adx(X')))) -> A_{_TL}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(tl(tl(nats))) -> A_{_TL}(a_{_tl}(a_{_nats}))
MARK(tl(hd(0))) -> A_{_TL}(a_{_hd}(0))
MARK(tl(hd(cons(X1', X2')))) -> A_{_TL}(a_{_hd}(cons(X1', X2')))
MARK(tl(hd(tl(X')))) -> A_{_TL}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(tl(hd(hd(X')))) -> A_{_TL}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(tl(hd(incr(X')))) -> A_{_TL}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(tl(hd(zeros))) -> A_{_TL}(a_{_hd}(a_{_zeros}))
MARK(tl(hd(adx(X')))) -> A_{_TL}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(tl(hd(nats))) -> A_{_TL}(a_{_hd}(a_{_nats}))
MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)
MARK(tl(hd(s(X')))) -> A_{_TL}(a_{_hd}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

The following dependency pairs can be strictly oriented:

MARK(tl(tl(s(X')))) -> A_{_TL}(a_{_tl}(s(X')))
MARK(tl(tl(0))) -> A_{_TL}(a_{_tl}(0))
MARK(tl(tl(cons(X1', X2')))) -> A_{_TL}(a_{_tl}(cons(X1', X2')))
MARK(tl(tl(tl(X')))) -> A_{_TL}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(tl(tl(hd(X')))) -> A_{_TL}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(tl(tl(incr(X')))) -> A_{_TL}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(tl(tl(zeros))) -> A_{_TL}(a_{_tl}(a_{_zeros}))
MARK(tl(tl(adx(X')))) -> A_{_TL}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(tl(tl(nats))) -> A_{_TL}(a_{_tl}(a_{_nats}))
MARK(tl(hd(0))) -> A_{_TL}(a_{_hd}(0))
MARK(tl(hd(cons(X1', X2')))) -> A_{_TL}(a_{_hd}(cons(X1', X2')))
MARK(tl(hd(tl(X')))) -> A_{_TL}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(tl(hd(hd(X')))) -> A_{_TL}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(tl(hd(incr(X')))) -> A_{_TL}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(tl(hd(zeros))) -> A_{_TL}(a_{_hd}(a_{_zeros}))
MARK(tl(hd(adx(X')))) -> A_{_TL}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(tl(hd(nats))) -> A_{_TL}(a_{_hd}(a_{_nats}))
MARK(tl(incr(s(X')))) -> A_{_TL}(a_{_incr}(s(X')))
MARK(tl(incr(0))) -> A_{_TL}(a_{_incr}(0))
MARK(tl(incr(cons(X1', X2')))) -> A_{_TL}(a_{_incr}(cons(X1', X2')))
MARK(tl(incr(tl(X')))) -> A_{_TL}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(tl(incr(hd(X')))) -> A_{_TL}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(tl(incr(incr(X')))) -> A_{_TL}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(tl(incr(zeros))) -> A_{_TL}(a_{_incr}(a_{_zeros}))
MARK(tl(incr(adx(X')))) -> A_{_TL}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(tl(incr(nats))) -> A_{_TL}(a_{_incr}(a_{_nats}))
MARK(tl(zeros)) -> A_{_TL}(cons(0, zeros))
MARK(tl(adx(s(X')))) -> A_{_TL}(a_{_adx}(s(X')))
MARK(tl(adx(0))) -> A_{_TL}(a_{_adx}(0))
MARK(tl(adx(cons(X1', X2')))) -> A_{_TL}(a_{_adx}(cons(X1', X2')))
MARK(tl(adx(tl(X')))) -> A_{_TL}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(tl(adx(hd(X')))) -> A_{_TL}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(tl(adx(incr(X')))) -> A_{_TL}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(tl(adx(zeros))) -> A_{_TL}(a_{_adx}(a_{_zeros}))
MARK(tl(adx(adx(X')))) -> A_{_TL}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(tl(adx(nats))) -> A_{_TL}(a_{_adx}(a_{_nats}))
MARK(tl(cons(X1', X2'))) -> A_{_TL}(cons(X1', X2'))
MARK(tl(X)) -> MARK(X)
MARK(tl(hd(s(X')))) -> A_{_TL}(a_{_hd}(s(X')))

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

a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(a__nats) = 0

POL(MARK(x₁)) = x₁

POL(adx(x₁)) = x₁

POL(a__zeros) = 0

POL(incr(x₁)) = x₁

POL(A__TL(x₁)) = x₁

POL(a__hd(x₁)) = x₁

POL(a__tl(x₁)) = 1 + x₁

POL(mark(x₁)) = x₁

POL(tl(x₁)) = 1 + x₁

POL(a__adx(x₁)) = x₁

POL(A__HD(x₁)) = x₁

POL(0) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(hd(x₁)) = x₁

POL(nats) = 0

POL(s(x₁)) = 0

POL(zeros) = 0

POL(a__incr(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 18

                 ↳Dependency Graph

Dependency Pairs:

MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))
MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
A_{_TL}(cons(X, Y)) -> MARK(Y)

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 19

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

The following dependency pairs can be strictly oriented:

MARK(hd(tl(0))) -> A_{_HD}(a_{_tl}(0))
MARK(hd(tl(cons(X1', X2')))) -> A_{_HD}(a_{_tl}(cons(X1', X2')))
MARK(hd(tl(tl(X')))) -> A_{_HD}(a_{_tl}(a_{_tl}(mark(X'))))
MARK(hd(tl(hd(X')))) -> A_{_HD}(a_{_tl}(a_{_hd}(mark(X'))))
MARK(hd(tl(incr(X')))) -> A_{_HD}(a_{_tl}(a_{_incr}(mark(X'))))
MARK(hd(tl(zeros))) -> A_{_HD}(a_{_tl}(a_{_zeros}))
MARK(hd(tl(adx(X')))) -> A_{_HD}(a_{_tl}(a_{_adx}(mark(X'))))
MARK(hd(tl(nats))) -> A_{_HD}(a_{_tl}(a_{_nats}))
MARK(hd(hd(s(X')))) -> A_{_HD}(a_{_hd}(s(X')))
MARK(hd(hd(0))) -> A_{_HD}(a_{_hd}(0))
MARK(hd(hd(cons(X1', X2')))) -> A_{_HD}(a_{_hd}(cons(X1', X2')))
MARK(hd(hd(tl(X')))) -> A_{_HD}(a_{_hd}(a_{_tl}(mark(X'))))
MARK(hd(hd(hd(X')))) -> A_{_HD}(a_{_hd}(a_{_hd}(mark(X'))))
MARK(hd(hd(incr(X')))) -> A_{_HD}(a_{_hd}(a_{_incr}(mark(X'))))
MARK(hd(hd(zeros))) -> A_{_HD}(a_{_hd}(a_{_zeros}))
MARK(hd(hd(adx(X')))) -> A_{_HD}(a_{_hd}(a_{_adx}(mark(X'))))
MARK(hd(hd(nats))) -> A_{_HD}(a_{_hd}(a_{_nats}))
MARK(hd(incr(s(X')))) -> A_{_HD}(a_{_incr}(s(X')))
MARK(hd(incr(0))) -> A_{_HD}(a_{_incr}(0))
MARK(hd(incr(cons(X1', X2')))) -> A_{_HD}(a_{_incr}(cons(X1', X2')))
MARK(hd(incr(tl(X')))) -> A_{_HD}(a_{_incr}(a_{_tl}(mark(X'))))
MARK(hd(incr(hd(X')))) -> A_{_HD}(a_{_incr}(a_{_hd}(mark(X'))))
MARK(hd(incr(incr(X')))) -> A_{_HD}(a_{_incr}(a_{_incr}(mark(X'))))
MARK(hd(incr(zeros))) -> A_{_HD}(a_{_incr}(a_{_zeros}))
MARK(hd(incr(adx(X')))) -> A_{_HD}(a_{_incr}(a_{_adx}(mark(X'))))
MARK(hd(incr(nats))) -> A_{_HD}(a_{_incr}(a_{_nats}))
MARK(hd(zeros)) -> A_{_HD}(cons(0, zeros))
MARK(hd(adx(s(X')))) -> A_{_HD}(a_{_adx}(s(X')))
MARK(hd(adx(0))) -> A_{_HD}(a_{_adx}(0))
MARK(hd(adx(cons(X1', X2')))) -> A_{_HD}(a_{_adx}(cons(X1', X2')))
MARK(hd(adx(tl(X')))) -> A_{_HD}(a_{_adx}(a_{_tl}(mark(X'))))
MARK(hd(adx(hd(X')))) -> A_{_HD}(a_{_adx}(a_{_hd}(mark(X'))))
MARK(hd(adx(incr(X')))) -> A_{_HD}(a_{_adx}(a_{_incr}(mark(X'))))
MARK(hd(adx(zeros))) -> A_{_HD}(a_{_adx}(a_{_zeros}))
MARK(hd(adx(adx(X')))) -> A_{_HD}(a_{_adx}(a_{_adx}(mark(X'))))
MARK(hd(adx(nats))) -> A_{_HD}(a_{_adx}(a_{_nats}))
MARK(hd(cons(X1', X2'))) -> A_{_HD}(cons(X1', X2'))
MARK(hd(X)) -> MARK(X)
MARK(hd(tl(s(X')))) -> A_{_HD}(a_{_tl}(s(X')))

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

a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(a__nats) = 0

POL(MARK(x₁)) = x₁

POL(adx(x₁)) = x₁

POL(a__zeros) = 0

POL(incr(x₁)) = x₁

POL(a__hd(x₁)) = 1 + x₁

POL(a__tl(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(tl(x₁)) = x₁

POL(a__adx(x₁)) = x₁

POL(A__HD(x₁)) = x₁

POL(0) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(hd(x₁)) = 1 + x₁

POL(nats) = 0

POL(s(x₁)) = 0

POL(zeros) = 0

POL(a__incr(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 20

                 ↳Dependency Graph

Dependency Pairs:

MARK(incr(X)) -> MARK(X)
A_{_HD}(cons(X, Y)) -> MARK(X)

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 21

                 ↳Polynomial Ordering

Dependency Pair:

MARK(incr(X)) -> MARK(X)

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

The following dependency pair can be strictly oriented:

MARK(incr(X)) -> MARK(X)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(incr(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 22

                 ↳Dependency Graph

Dependency Pair:

Rules:

a_{_nats} -> a_{_adx}(a_{_zeros})
a_{_nats} -> nats
a_{_zeros} -> cons(0, zeros)
a_{_zeros} -> zeros
a_{_incr}(cons(X, Y)) -> cons(s(X), incr(Y))
a_{_incr}(X) -> incr(X)
a_{_adx}(cons(X, Y)) -> a_{_incr}(cons(X, adx(Y)))
a_{_adx}(X) -> adx(X)
a_{_hd}(cons(X, Y)) -> mark(X)
a_{_hd}(X) -> hd(X)
a_{_tl}(cons(X, Y)) -> mark(Y)
a_{_tl}(X) -> tl(X)
mark(nats) -> a_{_nats}
mark(adx(X)) -> a_{_adx}(mark(X))
mark(zeros) -> a_{_zeros}
mark(incr(X)) -> a_{_incr}(mark(X))
mark(hd(X)) -> a_{_hd}(mark(X))
mark(tl(X)) -> a_{_tl}(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:30 minutes

POL(a__nats)	= 1
POL(MARK(x₁))	= x₁
POL(adx(x₁))	= x₁
POL(a__zeros)	= 0
POL(incr(x₁))	= x₁
POL(A__TL(x₁))	= x₁
POL(a__hd(x₁))	= x₁
POL(a__tl(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(tl(x₁))	= x₁
POL(a__adx(x₁))	= x₁
POL(A__HD(x₁))	= x₁
POL(0)	= 0
POL(cons(x₁, x₂))	= x₁ + x₂
POL(hd(x₁))	= x₁
POL(nats)	= 1
POL(s(x₁))	= 0
POL(zeros)	= 0
POL(a__incr(x₁))	= x₁