Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z]
sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)
FROM(X) -> FROM(s(X))
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)
SEL1(0, cons(X, Z)) -> QUOTE(X)
FIRST1(s(X), cons(Y, Z)) -> QUOTE(Y)
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)
QUOTE(s(X)) -> QUOTE(X)
QUOTE(sel(X, Z)) -> SEL1(X, Z)
QUOTE1(cons(X, Z)) -> QUOTE(X)
QUOTE1(cons(X, Z)) -> QUOTE1(Z)
QUOTE1(first(X, Z)) -> FIRST1(X, Z)
UNQUOTE(s1(X)) -> UNQUOTE(X)
UNQUOTE1(cons1(X, Z)) -> FCONS(unquote(X), unquote1(Z))
UNQUOTE1(cons1(X, Z)) -> UNQUOTE(X)
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Furthermore, R contains nine SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

SEL(s(X), cons(Y, Z)) -> SEL(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

SEL(s(X), cons(Y, Z)) -> SEL(X, Z)

one new Dependency Pair is created:

SEL(s(s(X'')), cons(Y, cons(Y'', Z''))) -> SEL(s(X''), cons(Y'', Z''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 10

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

SEL(s(s(X'')), cons(Y, cons(Y'', Z''))) -> SEL(s(X''), cons(Y'', Z''))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

SEL(s(s(X'')), cons(Y, cons(Y'', Z''))) -> SEL(s(X''), cons(Y'', Z''))

one new Dependency Pair is created:

SEL(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> SEL(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 10

             ↳FwdInst

...

               →DP Problem 11

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

SEL(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> SEL(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> SEL(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(SEL(x₁, x₂)) = x₁

POL(cons(x₁, x₂)) = 0

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 10

             ↳FwdInst

...

               →DP Problem 12

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

one new Dependency Pair is created:

FIRST(s(s(X'')), cons(Y, cons(Y'', Z''))) -> FIRST(s(X''), cons(Y'', Z''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 13

             ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FIRST(s(s(X'')), cons(Y, cons(Y'', Z''))) -> FIRST(s(X''), cons(Y'', Z''))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FIRST(s(s(X'')), cons(Y, cons(Y'', Z''))) -> FIRST(s(X''), cons(Y'', Z''))

one new Dependency Pair is created:

FIRST(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> FIRST(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 13

             ↳FwdInst

...

               →DP Problem 14

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FIRST(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> FIRST(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

The following dependency pair can be strictly oriented:

FIRST(s(s(s(X''''))), cons(Y, cons(Y''0, cons(Y'''', Z'''')))) -> FIRST(s(s(X'''')), cons(Y''0, cons(Y'''', Z'''')))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons(x₁, x₂)) = 0

POL(FIRST(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 13

             ↳FwdInst

...

               →DP Problem 15

                 ↳Dependency Graph

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Instantiation Transformation

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FROM(X) -> FROM(s(X))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FROM(X) -> FROM(s(X))

one new Dependency Pair is created:

FROM(s(X'')) -> FROM(s(s(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

           →DP Problem 16

             ↳Instantiation Transformation

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FROM(s(X'')) -> FROM(s(s(X'')))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FROM(s(X'')) -> FROM(s(s(X'')))

one new Dependency Pair is created:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

           →DP Problem 16

             ↳Inst

...

               →DP Problem 17

                 ↳Instantiation Transformation

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

one new Dependency Pair is created:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

           →DP Problem 16

             ↳Inst

...

               →DP Problem 18

                 ↳Instantiation Transformation

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

one new Dependency Pair is created:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

           →DP Problem 16

             ↳Inst

...

               →DP Problem 19

                 ↳Instantiation Transformation

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

       →DP Problem 9

         ↳Remaining

Dependency Pair:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:

innermost

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

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

one new Dependency Pair is created:

FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Inst

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

       →DP Problem 9

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE(s1(X)) -> UNQUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE(s(X)) -> QUOTE(X)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
UNQUOTE1(cons1(X, Z)) -> UNQUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
FIRST1(s(X), cons(Y, Z)) -> FIRST1(X, Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost
Dependency Pair:
QUOTE1(cons(X, Z)) -> QUOTE1(Z)

Rules:

sel(s(X), cons(Y, Z)) -> sel(X, Z)
sel(0, cons(X, Z)) -> X
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
from(X) -> cons(X, from(s(X)))
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
sel1(0, cons(X, Z)) -> quote(X)
first1(0, Z) -> nil1
first1(s(X), cons(Y, Z)) -> cons1(quote(Y), first1(X, Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(sel(X, Z)) -> sel1(X, Z)
quote1(cons(X, Z)) -> cons1(quote(X), quote1(Z))
quote1(nil) -> nil1
quote1(first(X, Z)) -> first1(X, Z)
unquote(01) -> 0
unquote(s1(X)) -> s(unquote(X))
unquote1(nil1) -> nil
unquote1(cons1(X, Z)) -> fcons(unquote(X), unquote1(Z))
fcons(X, Z) -> cons(X, Z)

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes

POL(SEL(x₁, x₂))	= x₁
POL(cons(x₁, x₂))	= 0
POL(s(x₁))	= 1 + x₁

POL(cons(x₁, x₂))	= 0
POL(FIRST(x₁, x₂))	= x₁
POL(s(x₁))	= 1 + x₁