Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
a_{_dbl}(0) -> 0
a_{_dbl}(s(X)) -> s(s(dbl(X)))
a_{_dbl}(X) -> dbl(X)
a_{_dbls}(nil) -> nil
a_{_dbls}(cons(X, Y)) -> cons(dbl(X), dbls(Y))
a_{_dbls}(X) -> dbls(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_indx}(nil, X) -> nil
a_{_indx}(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
a_{_indx}(X1, X2) -> indx(X1, X2)
a_{_from}(X) -> cons(X, from(s(X)))
a_{_from}(X) -> from(X)
a_{_dbl1}(0) -> 01
a_{_dbl1}(s(X)) -> s1(s1(a_{_dbl1}(mark(X))))
a_{_dbl1}(X) -> dbl1(X)
a_{_sel1}(0, cons(X, Y)) -> mark(X)
a_{_sel1}(s(X), cons(Y, Z)) -> a_{_sel1}(mark(X), mark(Z))
a_{_sel1}(X1, X2) -> sel1(X1, X2)
a_{_quote}(0) -> 01
a_{_quote}(s(X)) -> s1(a_{_quote}(mark(X)))
a_{_quote}(dbl(X)) -> a_{_dbl1}(mark(X))
a_{_quote}(sel(X, Y)) -> a_{_sel1}(mark(X), mark(Y))
a_{_quote}(X) -> quote(X)
mark(dbl(X)) -> a_{_dbl}(mark(X))
mark(dbls(X)) -> a_{_dbls}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(indx(X1, X2)) -> a_{_indx}(mark(X1), X2)
mark(from(X)) -> a_{_from}(X)
mark(dbl1(X)) -> a_{_dbl1}(mark(X))
mark(sel1(X1, X2)) -> a_{_sel1}(mark(X1), mark(X2))
mark(quote(X)) -> a_{_quote}(mark(X))
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)
mark(01) -> 01
mark(s1(X)) -> s1(mark(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_SEL}(0, cons(X, Y)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_DBL1}(s(X)) -> A_{_DBL1}(mark(X))
A_{_DBL1}(s(X)) -> MARK(X)
A_{_SEL1}(0, cons(X, Y)) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> A_{_SEL1}(mark(X), mark(Z))
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_QUOTE}(s(X)) -> A_{_QUOTE}(mark(X))
A_{_QUOTE}(s(X)) -> MARK(X)
A_{_QUOTE}(dbl(X)) -> A_{_DBL1}(mark(X))
A_{_QUOTE}(dbl(X)) -> MARK(X)
A_{_QUOTE}(sel(X, Y)) -> A_{_SEL1}(mark(X), mark(Y))
A_{_QUOTE}(sel(X, Y)) -> MARK(X)
A_{_QUOTE}(sel(X, Y)) -> MARK(Y)
MARK(dbl(X)) -> A_{_DBL}(mark(X))
MARK(dbl(X)) -> MARK(X)
MARK(dbls(X)) -> A_{_DBLS}(mark(X))
MARK(dbls(X)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(indx(X1, X2)) -> A_{_INDX}(mark(X1), X2)
MARK(indx(X1, X2)) -> MARK(X1)
MARK(from(X)) -> A_{_FROM}(X)
MARK(dbl1(X)) -> A_{_DBL1}(mark(X))
MARK(dbl1(X)) -> MARK(X)
MARK(sel1(X1, X2)) -> A_{_SEL1}(mark(X1), mark(X2))
MARK(sel1(X1, X2)) -> MARK(X1)
MARK(sel1(X1, X2)) -> MARK(X2)
MARK(quote(X)) -> A_{_QUOTE}(mark(X))
MARK(quote(X)) -> MARK(X)
MARK(s1(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

A_{_SEL}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_QUOTE}(sel(X, Y)) -> MARK(Y)
A_{_QUOTE}(sel(X, Y)) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(Z)
A_{_SEL1}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL1}(s(X), cons(Y, Z)) -> A_{_SEL1}(mark(X), mark(Z))
A_{_QUOTE}(sel(X, Y)) -> A_{_SEL1}(mark(X), mark(Y))
A_{_QUOTE}(dbl(X)) -> MARK(X)
A_{_QUOTE}(dbl(X)) -> A_{_DBL1}(mark(X))
MARK(s1(X)) -> MARK(X)
MARK(quote(X)) -> MARK(X)
A_{_QUOTE}(s(X)) -> MARK(X)
A_{_QUOTE}(s(X)) -> A_{_QUOTE}(mark(X))
MARK(quote(X)) -> A_{_QUOTE}(mark(X))
MARK(sel1(X1, X2)) -> MARK(X2)
MARK(sel1(X1, X2)) -> MARK(X1)
A_{_SEL1}(0, cons(X, Y)) -> MARK(X)
MARK(sel1(X1, X2)) -> A_{_SEL1}(mark(X1), mark(X2))
MARK(dbl1(X)) -> MARK(X)
A_{_DBL1}(s(X)) -> MARK(X)
A_{_DBL1}(s(X)) -> A_{_DBL1}(mark(X))
MARK(dbl1(X)) -> A_{_DBL1}(mark(X))
MARK(indx(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(dbls(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
A_{_SEL}(0, cons(X, Y)) -> MARK(X)

Rules:

a_{_dbl}(0) -> 0
a_{_dbl}(s(X)) -> s(s(dbl(X)))
a_{_dbl}(X) -> dbl(X)
a_{_dbls}(nil) -> nil
a_{_dbls}(cons(X, Y)) -> cons(dbl(X), dbls(Y))
a_{_dbls}(X) -> dbls(X)
a_{_sel}(0, cons(X, Y)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
a_{_indx}(nil, X) -> nil
a_{_indx}(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
a_{_indx}(X1, X2) -> indx(X1, X2)
a_{_from}(X) -> cons(X, from(s(X)))
a_{_from}(X) -> from(X)
a_{_dbl1}(0) -> 01
a_{_dbl1}(s(X)) -> s1(s1(a_{_dbl1}(mark(X))))
a_{_dbl1}(X) -> dbl1(X)
a_{_sel1}(0, cons(X, Y)) -> mark(X)
a_{_sel1}(s(X), cons(Y, Z)) -> a_{_sel1}(mark(X), mark(Z))
a_{_sel1}(X1, X2) -> sel1(X1, X2)
a_{_quote}(0) -> 01
a_{_quote}(s(X)) -> s1(a_{_quote}(mark(X)))
a_{_quote}(dbl(X)) -> a_{_dbl1}(mark(X))
a_{_quote}(sel(X, Y)) -> a_{_sel1}(mark(X), mark(Y))
a_{_quote}(X) -> quote(X)
mark(dbl(X)) -> a_{_dbl}(mark(X))
mark(dbls(X)) -> a_{_dbls}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(indx(X1, X2)) -> a_{_indx}(mark(X1), X2)
mark(from(X)) -> a_{_from}(X)
mark(dbl1(X)) -> a_{_dbl1}(mark(X))
mark(sel1(X1, X2)) -> a_{_sel1}(mark(X1), mark(X2))
mark(quote(X)) -> a_{_quote}(mark(X))
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)
mark(01) -> 01
mark(s1(X)) -> s1(mark(X))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:42 minutes