Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z]
dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

DBL(s(X)) -> DBL(X)
DBLS(cons(X, Y)) -> DBL(X)
DBLS(cons(X, Y)) -> DBLS(Y)
SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
INDX(cons(X, Y), Z) -> SEL(X, Z)
INDX(cons(X, Y), Z) -> INDX(Y, Z)
FROM(X) -> FROM(s(X))
DBL1(s(X)) -> DBL1(X)
SEL1(s(X), cons(Y, Z)) -> SEL1(X, Z)
QUOTE(s(X)) -> QUOTE(X)
QUOTE(dbl(X)) -> DBL1(X)
QUOTE(sel(X, Y)) -> SEL1(X, Y)

Furthermore, R contains eight 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

Dependency Pair:

DBL(s(X)) -> DBL(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:

innermost

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

DBL(s(X)) -> DBL(X)

one new Dependency Pair is created:

DBL(s(s(X''))) -> DBL(s(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 9

             ↳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

Dependency Pair:

DBL(s(s(X''))) -> DBL(s(X''))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:

innermost

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

DBL(s(s(X''))) -> DBL(s(X''))

one new Dependency Pair is created:

DBL(s(s(s(X'''')))) -> DBL(s(s(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 9

             ↳FwdInst

...

               →DP Problem 10

                 ↳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

Dependency Pair:

DBL(s(s(s(X'''')))) -> DBL(s(s(X'''')))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

DBL(s(s(s(X'''')))) -> DBL(s(s(X'''')))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(s(x₁)) = 1 + x₁

POL(DBL(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 9

             ↳FwdInst

...

               →DP Problem 11

                 ↳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

Dependency Pair:

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 2

         ↳FwdInst

           →DP Problem 12

             ↳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

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 2

         ↳FwdInst

           →DP Problem 12

             ↳FwdInst

...

               →DP Problem 13

                 ↳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

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:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 2

         ↳FwdInst

           →DP Problem 12

             ↳FwdInst

...

               →DP Problem 14

                 ↳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

Dependency Pair:

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 15

             ↳Instantiation Transformation

       →DP Problem 4

         ↳Remaining

       →DP Problem 5

         ↳Remaining

       →DP Problem 6

         ↳Remaining

       →DP Problem 7

         ↳Remaining

       →DP Problem 8

         ↳Remaining

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 15

             ↳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

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 15

             ↳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

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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 15

             ↳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

Dependency Pair:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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)

The following remains to be proven:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBL1(s(X)) -> DBL1(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBLS(cons(X, Y)) -> DBLS(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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)

The following remains to be proven:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBL1(s(X)) -> DBL1(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBLS(cons(X, Y)) -> DBLS(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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)

The following remains to be proven:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBL1(s(X)) -> DBL1(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBLS(cons(X, Y)) -> DBLS(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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)

The following remains to be proven:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBL1(s(X)) -> DBL1(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBLS(cons(X, Y)) -> DBLS(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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)

The following remains to be proven:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBL1(s(X)) -> DBL1(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBLS(cons(X, Y)) -> DBLS(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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)

The following remains to be proven:

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBL1(s(X)) -> DBL1(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

Strategy:
innermost
Dependency Pair:
DBLS(cons(X, Y)) -> DBLS(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, Z)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
from(X) -> cons(X, from(s(X)))
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) -> X
sel1(s(X), cons(Y, Z)) -> sel1(X, Z)
quote(0) -> 01
quote(s(X)) -> s1(quote(X))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)

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