Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(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)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_F}(X) -> MARK(X)
A_{_G}(s(X)) -> A_{_G}(mark(X))
A_{_G}(s(X)) -> MARK(X)
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)
MARK(f(X)) -> A_{_F}(mark(X))
MARK(f(X)) -> MARK(X)
MARK(g(X)) -> A_{_G}(mark(X))
MARK(g(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(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, Y)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(g(X)) -> MARK(X)
A_{_G}(s(X)) -> MARK(X)
A_{_G}(s(X)) -> A_{_G}(mark(X))
MARK(g(X)) -> A_{_G}(mark(X))
MARK(f(X)) -> MARK(X)
MARK(f(X)) -> A_{_F}(mark(X))
A_{_F}(X) -> MARK(X)

Rules:

a_{_f}(X) -> cons(mark(X), f(g(X)))
a_{_f}(X) -> f(X)
a_{_g}(0) -> s(0)
a_{_g}(s(X)) -> s(s(a_{_g}(mark(X))))
a_{_g}(X) -> g(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)
mark(f(X)) -> a_{_f}(mark(X))
mark(g(X)) -> a_{_g}(mark(X))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:05 minutes