Term Rewriting System R:
[X, Y, Z, X1, X2]
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ASEL(0, cons(X, Y)) -> MARK(X)
ASEL(s(X), cons(Y, Z)) -> ASEL(mark(X), mark(Z))
ASEL(s(X), cons(Y, Z)) -> MARK(X)
ASEL(s(X), cons(Y, Z)) -> MARK(Z)
MARK(dbl(X)) -> ADBL(mark(X))
MARK(dbl(X)) -> MARK(X)
MARK(dbls(X)) -> ADBLS(mark(X))
MARK(dbls(X)) -> MARK(X)
MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(indx(X1, X2)) -> AINDX(mark(X1), X2)
MARK(indx(X1, X2)) -> MARK(X1)
MARK(from(X)) -> AFROM(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

ASEL(s(X), cons(Y, Z)) -> MARK(Z)
MARK(indx(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
ASEL(s(X), cons(Y, Z)) -> MARK(X)
ASEL(s(X), cons(Y, Z)) -> ASEL(mark(X), mark(Z))
MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
MARK(dbls(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
ASEL(0, cons(X, Y)) -> MARK(X)


Rules:


adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)





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

ASEL(s(X), cons(Y, Z)) -> ASEL(mark(X), mark(Z))
18 new Dependency Pairs are created:

ASEL(s(dbl(X'')), cons(Y, Z)) -> ASEL(adbl(mark(X'')), mark(Z))
ASEL(s(dbls(X'')), cons(Y, Z)) -> ASEL(adbls(mark(X'')), mark(Z))
ASEL(s(sel(X1', X2')), cons(Y, Z)) -> ASEL(asel(mark(X1'), mark(X2')), mark(Z))
ASEL(s(indx(X1', X2')), cons(Y, Z)) -> ASEL(aindx(mark(X1'), X2'), mark(Z))
ASEL(s(from(X'')), cons(Y, Z)) -> ASEL(afrom(X''), mark(Z))
ASEL(s(0), cons(Y, Z)) -> ASEL(0, mark(Z))
ASEL(s(s(X'')), cons(Y, Z)) -> ASEL(s(X''), mark(Z))
ASEL(s(nil), cons(Y, Z)) -> ASEL(nil, mark(Z))
ASEL(s(cons(X1', X2')), cons(Y, Z)) -> ASEL(cons(X1', X2'), mark(Z))
ASEL(s(X), cons(Y, dbl(X''))) -> ASEL(mark(X), adbl(mark(X'')))
ASEL(s(X), cons(Y, dbls(X''))) -> ASEL(mark(X), adbls(mark(X'')))
ASEL(s(X), cons(Y, sel(X1', X2'))) -> ASEL(mark(X), asel(mark(X1'), mark(X2')))
ASEL(s(X), cons(Y, indx(X1', X2'))) -> ASEL(mark(X), aindx(mark(X1'), X2'))
ASEL(s(X), cons(Y, from(X''))) -> ASEL(mark(X), afrom(X''))
ASEL(s(X), cons(Y, 0)) -> ASEL(mark(X), 0)
ASEL(s(X), cons(Y, s(X''))) -> ASEL(mark(X), s(X''))
ASEL(s(X), cons(Y, nil)) -> ASEL(mark(X), nil)
ASEL(s(X), cons(Y, cons(X1', X2'))) -> ASEL(mark(X), cons(X1', X2'))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Narrowing Transformation


Dependency Pairs:

ASEL(s(X), cons(Y, cons(X1', X2'))) -> ASEL(mark(X), cons(X1', X2'))
ASEL(s(X), cons(Y, from(X''))) -> ASEL(mark(X), afrom(X''))
ASEL(s(X), cons(Y, indx(X1', X2'))) -> ASEL(mark(X), aindx(mark(X1'), X2'))
ASEL(s(X), cons(Y, sel(X1', X2'))) -> ASEL(mark(X), asel(mark(X1'), mark(X2')))
ASEL(s(X), cons(Y, dbls(X''))) -> ASEL(mark(X), adbls(mark(X'')))
ASEL(s(X), cons(Y, dbl(X''))) -> ASEL(mark(X), adbl(mark(X'')))
ASEL(s(s(X'')), cons(Y, Z)) -> ASEL(s(X''), mark(Z))
ASEL(s(0), cons(Y, Z)) -> ASEL(0, mark(Z))
ASEL(s(from(X'')), cons(Y, Z)) -> ASEL(afrom(X''), mark(Z))
ASEL(s(indx(X1', X2')), cons(Y, Z)) -> ASEL(aindx(mark(X1'), X2'), mark(Z))
ASEL(s(sel(X1', X2')), cons(Y, Z)) -> ASEL(asel(mark(X1'), mark(X2')), mark(Z))
ASEL(s(dbls(X'')), cons(Y, Z)) -> ASEL(adbls(mark(X'')), mark(Z))
ASEL(s(dbl(X'')), cons(Y, Z)) -> ASEL(adbl(mark(X'')), mark(Z))
ASEL(s(X), cons(Y, Z)) -> MARK(X)
MARK(indx(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
ASEL(0, cons(X, Y)) -> MARK(X)
MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
MARK(dbls(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
ASEL(s(X), cons(Y, Z)) -> MARK(Z)


Rules:


adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)





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

MARK(sel(X1, X2)) -> ASEL(mark(X1), mark(X2))
18 new Dependency Pairs are created:

MARK(sel(dbl(X'), X2)) -> ASEL(adbl(mark(X')), mark(X2))
MARK(sel(dbls(X'), X2)) -> ASEL(adbls(mark(X')), mark(X2))
MARK(sel(sel(X1'', X2''), X2)) -> ASEL(asel(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(indx(X1'', X2''), X2)) -> ASEL(aindx(mark(X1''), X2''), mark(X2))
MARK(sel(from(X'), X2)) -> ASEL(afrom(X'), mark(X2))
MARK(sel(0, X2)) -> ASEL(0, mark(X2))
MARK(sel(s(X'), X2)) -> ASEL(s(X'), mark(X2))
MARK(sel(nil, X2)) -> ASEL(nil, mark(X2))
MARK(sel(cons(X1'', X2''), X2)) -> ASEL(cons(X1'', X2''), mark(X2))
MARK(sel(X1, dbl(X'))) -> ASEL(mark(X1), adbl(mark(X')))
MARK(sel(X1, dbls(X'))) -> ASEL(mark(X1), adbls(mark(X')))
MARK(sel(X1, sel(X1'', X2''))) -> ASEL(mark(X1), asel(mark(X1''), mark(X2'')))
MARK(sel(X1, indx(X1'', X2''))) -> ASEL(mark(X1), aindx(mark(X1''), X2''))
MARK(sel(X1, from(X'))) -> ASEL(mark(X1), afrom(X'))
MARK(sel(X1, 0)) -> ASEL(mark(X1), 0)
MARK(sel(X1, s(X'))) -> ASEL(mark(X1), s(X'))
MARK(sel(X1, nil)) -> ASEL(mark(X1), nil)
MARK(sel(X1, cons(X1'', X2''))) -> ASEL(mark(X1), cons(X1'', X2''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Negative Polynomial Order


Dependency Pairs:

MARK(sel(X1, cons(X1'', X2''))) -> ASEL(mark(X1), cons(X1'', X2''))
MARK(sel(X1, from(X'))) -> ASEL(mark(X1), afrom(X'))
MARK(sel(X1, indx(X1'', X2''))) -> ASEL(mark(X1), aindx(mark(X1''), X2''))
MARK(sel(X1, sel(X1'', X2''))) -> ASEL(mark(X1), asel(mark(X1''), mark(X2'')))
MARK(sel(X1, dbls(X'))) -> ASEL(mark(X1), adbls(mark(X')))
MARK(sel(X1, dbl(X'))) -> ASEL(mark(X1), adbl(mark(X')))
MARK(sel(s(X'), X2)) -> ASEL(s(X'), mark(X2))
MARK(sel(0, X2)) -> ASEL(0, mark(X2))
MARK(sel(from(X'), X2)) -> ASEL(afrom(X'), mark(X2))
MARK(sel(indx(X1'', X2''), X2)) -> ASEL(aindx(mark(X1''), X2''), mark(X2))
ASEL(s(X), cons(Y, from(X''))) -> ASEL(mark(X), afrom(X''))
ASEL(s(X), cons(Y, indx(X1', X2'))) -> ASEL(mark(X), aindx(mark(X1'), X2'))
ASEL(s(X), cons(Y, sel(X1', X2'))) -> ASEL(mark(X), asel(mark(X1'), mark(X2')))
ASEL(s(X), cons(Y, dbls(X''))) -> ASEL(mark(X), adbls(mark(X'')))
ASEL(s(X), cons(Y, dbl(X''))) -> ASEL(mark(X), adbl(mark(X'')))
ASEL(s(s(X'')), cons(Y, Z)) -> ASEL(s(X''), mark(Z))
ASEL(s(0), cons(Y, Z)) -> ASEL(0, mark(Z))
ASEL(s(from(X'')), cons(Y, Z)) -> ASEL(afrom(X''), mark(Z))
ASEL(s(indx(X1', X2')), cons(Y, Z)) -> ASEL(aindx(mark(X1'), X2'), mark(Z))
ASEL(s(sel(X1', X2')), cons(Y, Z)) -> ASEL(asel(mark(X1'), mark(X2')), mark(Z))
ASEL(s(dbls(X'')), cons(Y, Z)) -> ASEL(adbls(mark(X'')), mark(Z))
ASEL(s(dbl(X'')), cons(Y, Z)) -> ASEL(adbl(mark(X'')), mark(Z))
MARK(sel(sel(X1'', X2''), X2)) -> ASEL(asel(mark(X1''), mark(X2'')), mark(X2))
ASEL(s(X), cons(Y, Z)) -> MARK(Z)
MARK(sel(dbls(X'), X2)) -> ASEL(adbls(mark(X')), mark(X2))
ASEL(s(X), cons(Y, Z)) -> MARK(X)
MARK(sel(dbl(X'), X2)) -> ASEL(adbl(mark(X')), mark(X2))
MARK(indx(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
MARK(dbls(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
ASEL(0, cons(X, Y)) -> MARK(X)
ASEL(s(X), cons(Y, cons(X1', X2'))) -> ASEL(mark(X), cons(X1', X2'))


Rules:


adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)





The following Dependency Pairs can be strictly oriented using the given order.

MARK(sel(X1, dbl(X'))) -> ASEL(mark(X1), adbl(mark(X')))
ASEL(s(X), cons(Y, dbl(X''))) -> ASEL(mark(X), adbl(mark(X'')))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)


Used ordering:
Polynomial Order with Interpretation:

POL( MARK(x1) ) = 1

POL( ASEL(x1, x2) ) = x2

POL( adbl(x1) ) = 0

POL( cons(x1, x2) ) = 1

POL( mark(x1) ) = 1

POL( asel(x1, x2) ) = 1

POL( afrom(x1) ) = 1

POL( aindx(x1, x2) ) = 1

POL( adbls(x1) ) = 1

POL( 0 ) = 0

POL( s(x1) ) = 0

POL( nil ) = 0

POL( from(x1) ) = 0

POL( indx(x1, x2) ) = 0

POL( sel(x1, x2) ) = 0

POL( dbls(x1) ) = 0

POL( dbl(x1) ) = 0


This results in one new DP problem.


   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 4
Negative Polynomial Order


Dependency Pairs:

MARK(sel(X1, cons(X1'', X2''))) -> ASEL(mark(X1), cons(X1'', X2''))
MARK(sel(X1, from(X'))) -> ASEL(mark(X1), afrom(X'))
MARK(sel(X1, indx(X1'', X2''))) -> ASEL(mark(X1), aindx(mark(X1''), X2''))
MARK(sel(X1, sel(X1'', X2''))) -> ASEL(mark(X1), asel(mark(X1''), mark(X2'')))
MARK(sel(X1, dbls(X'))) -> ASEL(mark(X1), adbls(mark(X')))
MARK(sel(s(X'), X2)) -> ASEL(s(X'), mark(X2))
MARK(sel(0, X2)) -> ASEL(0, mark(X2))
MARK(sel(from(X'), X2)) -> ASEL(afrom(X'), mark(X2))
MARK(sel(indx(X1'', X2''), X2)) -> ASEL(aindx(mark(X1''), X2''), mark(X2))
ASEL(s(X), cons(Y, from(X''))) -> ASEL(mark(X), afrom(X''))
ASEL(s(X), cons(Y, indx(X1', X2'))) -> ASEL(mark(X), aindx(mark(X1'), X2'))
ASEL(s(X), cons(Y, sel(X1', X2'))) -> ASEL(mark(X), asel(mark(X1'), mark(X2')))
ASEL(s(X), cons(Y, dbls(X''))) -> ASEL(mark(X), adbls(mark(X'')))
ASEL(s(s(X'')), cons(Y, Z)) -> ASEL(s(X''), mark(Z))
ASEL(s(0), cons(Y, Z)) -> ASEL(0, mark(Z))
ASEL(s(from(X'')), cons(Y, Z)) -> ASEL(afrom(X''), mark(Z))
ASEL(s(indx(X1', X2')), cons(Y, Z)) -> ASEL(aindx(mark(X1'), X2'), mark(Z))
ASEL(s(sel(X1', X2')), cons(Y, Z)) -> ASEL(asel(mark(X1'), mark(X2')), mark(Z))
ASEL(s(dbls(X'')), cons(Y, Z)) -> ASEL(adbls(mark(X'')), mark(Z))
ASEL(s(dbl(X'')), cons(Y, Z)) -> ASEL(adbl(mark(X'')), mark(Z))
MARK(sel(sel(X1'', X2''), X2)) -> ASEL(asel(mark(X1''), mark(X2'')), mark(X2))
ASEL(s(X), cons(Y, Z)) -> MARK(Z)
MARK(sel(dbls(X'), X2)) -> ASEL(adbls(mark(X')), mark(X2))
ASEL(s(X), cons(Y, Z)) -> MARK(X)
MARK(sel(dbl(X'), X2)) -> ASEL(adbl(mark(X')), mark(X2))
MARK(indx(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
MARK(dbls(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
ASEL(0, cons(X, Y)) -> MARK(X)
ASEL(s(X), cons(Y, cons(X1', X2'))) -> ASEL(mark(X), cons(X1', X2'))


Rules:


adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)





The following Dependency Pairs can be strictly oriented using the given order.

MARK(sel(from(X'), X2)) -> ASEL(afrom(X'), mark(X2))
MARK(sel(indx(X1'', X2''), X2)) -> ASEL(aindx(mark(X1''), X2''), mark(X2))
ASEL(s(from(X'')), cons(Y, Z)) -> ASEL(afrom(X''), mark(Z))
ASEL(s(indx(X1', X2')), cons(Y, Z)) -> ASEL(aindx(mark(X1'), X2'), mark(Z))
ASEL(s(dbls(X'')), cons(Y, Z)) -> ASEL(adbls(mark(X'')), mark(Z))
MARK(sel(dbls(X'), X2)) -> ASEL(adbls(mark(X')), mark(X2))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)


Used ordering:
Polynomial Order with Interpretation:

POL( MARK(x1) ) = 1

POL( ASEL(x1, x2) ) = x1

POL( afrom(x1) ) = 0

POL( s(x1) ) = 1

POL( asel(x1, x2) ) = 1

POL( 0 ) = 1

POL( mark(x1) ) = 1

POL( adbls(x1) ) = 0

POL( adbl(x1) ) = 1

POL( aindx(x1, x2) ) = 0

POL( nil ) = 0

POL( cons(x1, x2) ) = 0

POL( from(x1) ) = 0

POL( indx(x1, x2) ) = 0

POL( sel(x1, x2) ) = 0

POL( dbls(x1) ) = 0

POL( dbl(x1) ) = 0


This results in one new DP problem.


   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 5
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

MARK(sel(X1, cons(X1'', X2''))) -> ASEL(mark(X1), cons(X1'', X2''))
MARK(sel(X1, from(X'))) -> ASEL(mark(X1), afrom(X'))
MARK(sel(X1, indx(X1'', X2''))) -> ASEL(mark(X1), aindx(mark(X1''), X2''))
MARK(sel(X1, sel(X1'', X2''))) -> ASEL(mark(X1), asel(mark(X1''), mark(X2'')))
MARK(sel(X1, dbls(X'))) -> ASEL(mark(X1), adbls(mark(X')))
MARK(sel(s(X'), X2)) -> ASEL(s(X'), mark(X2))
MARK(sel(0, X2)) -> ASEL(0, mark(X2))
ASEL(s(X), cons(Y, from(X''))) -> ASEL(mark(X), afrom(X''))
ASEL(s(X), cons(Y, indx(X1', X2'))) -> ASEL(mark(X), aindx(mark(X1'), X2'))
ASEL(s(X), cons(Y, sel(X1', X2'))) -> ASEL(mark(X), asel(mark(X1'), mark(X2')))
ASEL(s(X), cons(Y, dbls(X''))) -> ASEL(mark(X), adbls(mark(X'')))
ASEL(s(s(X'')), cons(Y, Z)) -> ASEL(s(X''), mark(Z))
ASEL(s(0), cons(Y, Z)) -> ASEL(0, mark(Z))
ASEL(s(sel(X1', X2')), cons(Y, Z)) -> ASEL(asel(mark(X1'), mark(X2')), mark(Z))
ASEL(s(dbl(X'')), cons(Y, Z)) -> ASEL(adbl(mark(X'')), mark(Z))
MARK(sel(sel(X1'', X2''), X2)) -> ASEL(asel(mark(X1''), mark(X2'')), mark(X2))
ASEL(s(X), cons(Y, Z)) -> MARK(Z)
ASEL(s(X), cons(Y, Z)) -> MARK(X)
MARK(sel(dbl(X'), X2)) -> ASEL(adbl(mark(X')), mark(X2))
MARK(indx(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
MARK(dbls(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
ASEL(0, cons(X, Y)) -> MARK(X)
ASEL(s(X), cons(Y, cons(X1', X2'))) -> ASEL(mark(X), cons(X1', X2'))


Rules:


adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
adbls(nil) -> nil
adbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
adbls(X) -> dbls(X)
asel(0, cons(X, Y)) -> mark(X)
asel(s(X), cons(Y, Z)) -> asel(mark(X), mark(Z))
asel(X1, X2) -> sel(X1, X2)
aindx(nil, X) -> nil
aindx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
aindx(X1, X2) -> indx(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(dbl(X)) -> adbl(mark(X))
mark(dbls(X)) -> adbls(mark(X))
mark(sel(X1, X2)) -> asel(mark(X1), mark(X2))
mark(indx(X1, X2)) -> aindx(mark(X1), X2)
mark(from(X)) -> afrom(X)
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes