Termination Proof using AProVE

Term Rewriting System R:
[X, Z, Y, X1, X2]
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_first}(0, Z) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_sel}(0, cons(X, Z)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(first(X1, X2)) -> a_{_first}(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_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
A_{_SEL}(0, cons(X, Z)) -> 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)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(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(X), cons(Y, Z)) -> MARK(Z)
A_{_SEL}(s(X), cons(Y, Z)) -> MARK(X)
A_{_SEL}(s(X), cons(Y, Z)) -> A_{_SEL}(mark(X), mark(Z))
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, Z)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A_{_FIRST}(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> A_{_FIRST}(mark(X1), mark(X2))
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_{_first}(0, Z) -> nil
a_{_first}(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
a_{_first}(X1, X2) -> first(X1, X2)
a_{_sel}(0, cons(X, Z)) -> mark(X)
a_{_sel}(s(X), cons(Y, Z)) -> a_{_sel}(mark(X), mark(Z))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(first(X1, X2)) -> a_{_first}(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:
0:01 minutes