Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, X1, X2]
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_FROM}(X) -> MARK(X)
A_{_HEAD}(cons(X, XS)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> A_{_HEAD}(mark(XS))
A_{_2ND}(cons(X, XS)) -> MARK(XS)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(head(X)) -> A_{_HEAD}(mark(X))
MARK(head(X)) -> MARK(X)
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(2nd(X)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X1)
MARK(take(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(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(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(2nd(X)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> MARK(XS)
A_{_2ND}(cons(X, XS)) -> A_{_HEAD}(mark(XS))
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(head(X)) -> MARK(X)
A_{_HEAD}(cons(X, XS)) -> MARK(X)
MARK(head(X)) -> A_{_HEAD}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_FROM}(X) -> MARK(X)

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:09 minutes