Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
A_{_FROM}(X) -> MARK(X)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(s(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(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))

12 new Dependency Pairs are created:

MARK(first(first(X1'', X2''), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), mark(X2))
MARK(first(from(X'), X2)) -> A_{_FIRST}(a_{_from}(mark(X')), mark(X2))
MARK(first(0, X2)) -> A_{_FIRST}(0, mark(X2))
MARK(first(nil, X2)) -> A_{_FIRST}(nil, mark(X2))
MARK(first(s(X'), X2)) -> A_{_FIRST}(s(mark(X')), mark(X2))
MARK(first(cons(X1'', X2''), X2)) -> A_{_FIRST}(cons(mark(X1''), X2''), mark(X2))
MARK(first(X1, first(X1'', X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))
MARK(first(X1, 0)) -> A_{_FIRST}(mark(X1), 0)
MARK(first(X1, nil)) -> A_{_FIRST}(mark(X1), nil)
MARK(first(X1, s(X'))) -> A_{_FIRST}(mark(X1), s(mark(X')))
MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))
MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), X2)) -> A_{_FIRST}(s(mark(X')), mark(X2))
MARK(first(from(X'), X2)) -> A_{_FIRST}(a_{_from}(mark(X')), mark(X2))
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(first(X1'', X2''), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), mark(X2))
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(first(X1'', X2''), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), mark(X2))

19 new Dependency Pairs are created:

MARK(first(first(X1''', X2'''), X2)) -> A_{_FIRST}(first(mark(X1'''), mark(X2''')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), 0)) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), 0)
MARK(first(first(X1'', X2''), nil)) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), nil)
MARK(first(first(X1'', X2''), s(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), s(mark(X')))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))

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(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), X2)) -> A_{_FIRST}(s(mark(X')), mark(X2))
MARK(first(from(X'), X2)) -> A_{_FIRST}(a_{_from}(mark(X')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(from(X'), X2)) -> A_{_FIRST}(a_{_from}(mark(X')), mark(X2))

14 new Dependency Pairs are created:

MARK(first(from(X''), X2)) -> A_{_FIRST}(cons(mark(mark(X'')), from(s(mark(X'')))), mark(X2))
MARK(first(from(X''), X2)) -> A_{_FIRST}(from(mark(X'')), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), 0)) -> A_{_FIRST}(a_{_from}(mark(X')), 0)
MARK(first(from(X'), nil)) -> A_{_FIRST}(a_{_from}(mark(X')), nil)
MARK(first(from(X'), s(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), s(mark(X'')))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))

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(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))
MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), X2)) -> A_{_FIRST}(s(mark(X')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(s(X'), X2)) -> A_{_FIRST}(s(mark(X')), mark(X2))

12 new Dependency Pairs are created:

MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(0), X2)) -> A_{_FIRST}(s(0), mark(X2))
MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), 0)) -> A_{_FIRST}(s(mark(X')), 0)
MARK(first(s(X'), nil)) -> A_{_FIRST}(s(mark(X')), nil)
MARK(first(s(X'), s(X''))) -> A_{_FIRST}(s(mark(X')), s(mark(X'')))
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))

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(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(s(0), X2)) -> A_{_FIRST}(s(0), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))
MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), mark(X2'')))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(X1, first(X1'', X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), mark(X2'')))

19 new Dependency Pairs are created:

MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(0, first(X1'', X2''))) -> A_{_FIRST}(0, a_{_first}(mark(X1''), mark(X2'')))
MARK(first(nil, first(X1'', X2''))) -> A_{_FIRST}(nil, a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(cons(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(cons(mark(X1'''), X2'), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(X1, first(X1''', X2'''))) -> A_{_FIRST}(mark(X1), first(mark(X1'''), mark(X2''')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))

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(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(s(0), X2)) -> A_{_FIRST}(s(0), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))
MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(X1, from(X'))) -> A_{_FIRST}(mark(X1), a_{_from}(mark(X')))

14 new Dependency Pairs are created:

MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(0, from(X'))) -> A_{_FIRST}(0, a_{_from}(mark(X')))
MARK(first(nil, from(X'))) -> A_{_FIRST}(nil, a_{_from}(mark(X')))
MARK(first(s(X''), from(X'))) -> A_{_FIRST}(s(mark(X'')), a_{_from}(mark(X')))
MARK(first(cons(X1'', X2'), from(X'))) -> A_{_FIRST}(cons(mark(X1''), X2'), a_{_from}(mark(X')))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), from(mark(X'')))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))

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(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(s(X''), from(X'))) -> A_{_FIRST}(s(mark(X'')), a_{_from}(mark(X')))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(s(0), X2)) -> A_{_FIRST}(s(0), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

MARK(first(X1, cons(X1'', X2''))) -> A_{_FIRST}(mark(X1), cons(mark(X1''), X2''))

12 new Dependency Pairs are created:

MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(from(X'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1''), X2''))
MARK(first(0, cons(X1'', X2''))) -> A_{_FIRST}(0, cons(mark(X1''), X2''))
MARK(first(nil, cons(X1'', X2''))) -> A_{_FIRST}(nil, cons(mark(X1''), X2''))
MARK(first(s(X'), cons(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1''), X2''))
MARK(first(cons(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(cons(mark(X1'''), X2'), cons(mark(X1''), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))
MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(s(X'), cons(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1''), X2''))
MARK(first(from(X'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1''), X2''))
MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(s(X''), from(X'))) -> A_{_FIRST}(s(mark(X'')), a_{_from}(mark(X')))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(s(0), X2)) -> A_{_FIRST}(s(0), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

The following dependency pairs can be strictly oriented:

MARK(first(s(0), X2)) -> A_{_FIRST}(s(0), mark(X2))
MARK(first(from(0), X2)) -> A_{_FIRST}(a_{_from}(0), mark(X2))
MARK(first(first(X1'', 0), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), 0), mark(X2))
MARK(first(first(0, X2''), X2)) -> A_{_FIRST}(a_{_first}(0, mark(X2'')), mark(X2))

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

mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 1

POL(MARK(x₁)) = x₁

POL(A__FIRST(x₁, x₂)) = x₂

POL(cons(x₁, x₂)) = x₁

POL(nil) = 0

POL(A__FROM(x₁)) = x₁

POL(s(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(a__from(x₁)) = x₁

POL(a__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))
MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(s(X'), cons(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1''), X2''))
MARK(first(from(X'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1''), X2''))
MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(s(X''), from(X'))) -> A_{_FIRST}(s(mark(X'')), a_{_from}(mark(X')))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

The following dependency pairs can be strictly oriented:

MARK(first(s(nil), X2)) -> A_{_FIRST}(s(nil), mark(X2))
MARK(first(from(nil), X2)) -> A_{_FIRST}(a_{_from}(nil), mark(X2))
MARK(first(first(X1'', nil), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), nil), mark(X2))
MARK(first(first(nil, X2''), X2)) -> A_{_FIRST}(a_{_first}(nil, mark(X2'')), mark(X2))

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

mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 1

POL(MARK(x₁)) = x₁

POL(A__FIRST(x₁, x₂)) = x₂

POL(cons(x₁, x₂)) = x₁

POL(nil) = 1

POL(A__FROM(x₁)) = x₁

POL(s(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(a__from(x₁)) = x₁

POL(a__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))
MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(s(X'), cons(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1''), X2''))
MARK(first(from(X'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1''), X2''))
MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(s(X''), from(X'))) -> A_{_FIRST}(s(mark(X'')), a_{_from}(mark(X')))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

The following dependency pairs can be strictly oriented:

MARK(first(s(X'), cons(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1''), X2''))
MARK(first(s(X''), from(X'))) -> A_{_FIRST}(s(mark(X'')), a_{_from}(mark(X')))
MARK(first(s(X'), first(X1'', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(s(X'), cons(X1', X2''))) -> A_{_FIRST}(s(mark(X')), cons(mark(X1'), X2''))
MARK(first(s(X'), from(X''))) -> A_{_FIRST}(s(mark(X')), a_{_from}(mark(X'')))
MARK(first(s(X'), first(X1', X2''))) -> A_{_FIRST}(s(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(s(cons(X1', X2'')), X2)) -> A_{_FIRST}(s(cons(mark(X1'), X2'')), mark(X2))
MARK(first(s(s(X'')), X2)) -> A_{_FIRST}(s(s(mark(X''))), mark(X2))
MARK(first(s(from(X'')), X2)) -> A_{_FIRST}(s(a_{_from}(mark(X''))), mark(X2))
MARK(first(s(first(X1', X2'')), X2)) -> A_{_FIRST}(s(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(from(s(X'')), X2)) -> A_{_FIRST}(a_{_from}(s(mark(X''))), mark(X2))
MARK(first(first(X1'', s(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), s(mark(X'))), mark(X2))
MARK(first(first(s(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(s(mark(X')), mark(X2'')), mark(X2))
MARK(s(X)) -> MARK(X)

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

mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(MARK(x₁)) = x₁

POL(A__FIRST(x₁, x₂)) = x₂

POL(cons(x₁, x₂)) = x₁

POL(nil) = 0

POL(A__FROM(x₁)) = x₁

POL(s(x₁)) = 1 + x₁

POL(mark(x₁)) = x₁

POL(a__from(x₁)) = x₁

POL(a__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))
MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(from(X'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1''), X2''))
MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

The following dependency pairs can be strictly oriented:

MARK(first(from(X'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1''), X2''))
MARK(first(X1, from(X''))) -> A_{_FIRST}(mark(X1), cons(mark(mark(X'')), from(s(mark(X'')))))
MARK(first(from(X''), from(X'))) -> A_{_FIRST}(a_{_from}(mark(X'')), a_{_from}(mark(X')))
MARK(first(from(X'), first(X1'', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(from(X'), cons(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), cons(mark(X1'), X2''))
MARK(first(from(X'), from(X''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_from}(mark(X'')))
MARK(first(from(X'), first(X1', X2''))) -> A_{_FIRST}(a_{_from}(mark(X')), a_{_first}(mark(X1'), mark(X2'')))
MARK(first(from(cons(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(cons(mark(X1'), X2'')), mark(X2))
MARK(first(from(from(X'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_from}(mark(X''))), mark(X2))
MARK(first(from(first(X1', X2'')), X2)) -> A_{_FIRST}(a_{_from}(a_{_first}(mark(X1'), mark(X2''))), mark(X2))
MARK(first(first(X1'', from(X')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_from}(mark(X'))), mark(X2))
MARK(first(first(from(X'), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_from}(mark(X')), mark(X2'')), mark(X2))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))

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

mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(MARK(x₁)) = x₁

POL(A__FIRST(x₁, x₂)) = x₂

POL(cons(x₁, x₂)) = x₁

POL(nil) = 0

POL(A__FROM(x₁)) = x₁

POL(s(x₁)) = 0

POL(mark(x₁)) = x₁

POL(a__from(x₁)) = 1 + x₁

POL(a__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Dependency Graph

Dependency Pairs:

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

The following dependency pairs can be strictly oriented:

MARK(first(first(X1'', cons(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), cons(mark(X1'), X2''')), mark(X2))
MARK(first(first(cons(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(cons(mark(X1'), X2'''), mark(X2'')), mark(X2))
MARK(cons(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)

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

mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(first(x₁, x₂)) = x₁ + x₂

POL(0) = 0

POL(MARK(x₁)) = x₁

POL(A__FIRST(x₁, x₂)) = x₂

POL(cons(x₁, x₂)) = 1 + x₁

POL(nil) = 0

POL(s(x₁)) = 0

POL(mark(x₁)) = x₁

POL(a__from(x₁)) = 1 + x₁

POL(a__first(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Dependency Graph

Dependency Pairs:

MARK(first(X1, cons(s(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(s(mark(X')), X2''))
MARK(first(X1, cons(nil, X2''))) -> A_{_FIRST}(mark(X1), cons(nil, X2''))
MARK(first(X1, cons(0, X2''))) -> A_{_FIRST}(mark(X1), cons(0, X2''))
MARK(first(X1, cons(from(X'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_from}(mark(X')), X2''))
MARK(first(X1, cons(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(a_{_first}(mark(X1'''), mark(X2')), X2''))
MARK(first(first(X1''', X2'), cons(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), cons(mark(X1''), X2''))
MARK(first(X1, from(cons(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(cons(mark(X1''), X2')))
MARK(first(X1, from(s(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(s(mark(X''))))
MARK(first(X1, from(nil))) -> A_{_FIRST}(mark(X1), a_{_from}(nil))
MARK(first(X1, from(0))) -> A_{_FIRST}(mark(X1), a_{_from}(0))
MARK(first(X1, from(from(X'')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_from}(mark(X''))))
MARK(first(X1, from(first(X1'', X2')))) -> A_{_FIRST}(mark(X1), a_{_from}(a_{_first}(mark(X1''), mark(X2'))))
MARK(first(first(X1'', X2'), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2')), a_{_from}(mark(X')))
MARK(first(X1, first(X1'', cons(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), cons(mark(X1'''), X2')))
MARK(first(X1, first(X1'', s(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), s(mark(X'))))
MARK(first(X1, first(X1'', nil))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), nil))
MARK(first(X1, first(X1'', 0))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), 0))
MARK(first(X1, first(X1'', from(X')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_from}(mark(X'))))
MARK(first(X1, first(X1'', first(X1''', X2')))) -> A_{_FIRST}(mark(X1), a_{_first}(mark(X1''), a_{_first}(mark(X1'''), mark(X2'))))
MARK(first(X1, first(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(cons(mark(X1'''), X2'), mark(X2'')))
MARK(first(X1, first(s(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(s(mark(X')), mark(X2'')))
MARK(first(X1, first(nil, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(nil, mark(X2'')))
MARK(first(X1, first(0, X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(0, mark(X2'')))
MARK(first(X1, first(from(X'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_from}(mark(X')), mark(X2'')))
MARK(first(X1, first(first(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), a_{_first}(a_{_first}(mark(X1'''), mark(X2')), mark(X2'')))
MARK(first(first(X1''', X2'), first(X1'', X2''))) -> A_{_FIRST}(a_{_first}(mark(X1'''), mark(X2')), a_{_first}(mark(X1''), mark(X2'')))
MARK(first(first(X1'', X2''), cons(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), cons(mark(X1'), X2'''))
MARK(first(first(X1'', X2''), from(X'))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_from}(mark(X')))
MARK(first(first(X1'', X2''), first(X1', X2'''))) -> A_{_FIRST}(a_{_first}(mark(X1''), mark(X2'')), a_{_first}(mark(X1'), mark(X2''')))
MARK(first(first(X1'', first(X1', X2''')), X2)) -> A_{_FIRST}(a_{_first}(mark(X1''), a_{_first}(mark(X1'), mark(X2'''))), mark(X2))
MARK(first(first(first(X1', X2'''), X2''), X2)) -> A_{_FIRST}(a_{_first}(a_{_first}(mark(X1'), mark(X2''')), mark(X2'')), mark(X2))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, cons(cons(X1''', X2'), X2''))) -> A_{_FIRST}(mark(X1), cons(cons(mark(X1'''), X2'), X2''))

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

The following dependency pairs can be strictly oriented:

MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)

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(first(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Dependency Graph

Dependency Pair:

Rules:

a_{_first}(0, X) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
mark(first(X1, X2)) -> a_{_first}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(0) -> 0
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:38 minutes

POL(from(x₁))	= x₁
POL(first(x₁, x₂))	= x₁ + x₂
POL(0)	= 1
POL(MARK(x₁))	= x₁
POL(A__FIRST(x₁, x₂))	= x₂
POL(cons(x₁, x₂))	= x₁
POL(nil)	= 0
POL(A__FROM(x₁))	= x₁
POL(s(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(a__from(x₁))	= x₁
POL(a__first(x₁, x₂))	= x₁ + x₂

POL(from(x₁))	= 1 + x₁
POL(first(x₁, x₂))	= x₁ + x₂
POL(0)	= 0
POL(MARK(x₁))	= x₁
POL(A__FIRST(x₁, x₂))	= x₂
POL(cons(x₁, x₂))	= x₁
POL(nil)	= 0
POL(A__FROM(x₁))	= x₁
POL(s(x₁))	= 0
POL(mark(x₁))	= x₁
POL(a__from(x₁))	= 1 + x₁
POL(a__first(x₁, x₂))	= x₁ + x₂