Termination Proof using AProVE

Term Rewriting System R:
[YS, X, XS, X1, X2, Y, L]
a_{_app}(nil, YS) -> mark(YS)
a_{_app}(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
a_{_app}(X1, X2) -> app(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_zWadr}(nil, YS) -> nil
a_{_zWadr}(XS, nil) -> nil
a_{_zWadr}(cons(X, XS), cons(Y, YS)) -> cons(a_{_app}(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a_{_zWadr}(X1, X2) -> zWadr(X1, X2)
a_{_prefix}(L) -> cons(nil, zWadr(L, prefix(L)))
a_{_prefix}(X) -> prefix(X)
mark(app(X1, X2)) -> a_{_app}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(zWadr(X1, X2)) -> a_{_zWadr}(mark(X1), mark(X2))
mark(prefix(X)) -> a_{_prefix}(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_APP}(nil, YS) -> MARK(YS)
A_{_APP}(cons(X, XS), YS) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
A_{_ZWADR}(cons(X, XS), cons(Y, YS)) -> A_{_APP}(mark(Y), cons(mark(X), nil))
A_{_ZWADR}(cons(X, XS), cons(Y, YS)) -> MARK(Y)
A_{_ZWADR}(cons(X, XS), cons(Y, YS)) -> MARK(X)
MARK(app(X1, X2)) -> A_{_APP}(mark(X1), mark(X2))
MARK(app(X1, X2)) -> MARK(X1)
MARK(app(X1, X2)) -> MARK(X2)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(zWadr(X1, X2)) -> A_{_ZWADR}(mark(X1), mark(X2))
MARK(zWadr(X1, X2)) -> MARK(X1)
MARK(zWadr(X1, X2)) -> MARK(X2)
MARK(prefix(X)) -> A_{_PREFIX}(mark(X))
MARK(prefix(X)) -> MARK(X)
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_{_ZWADR}(cons(X, XS), cons(Y, YS)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(prefix(X)) -> MARK(X)
MARK(zWadr(X1, X2)) -> MARK(X2)
MARK(zWadr(X1, X2)) -> MARK(X1)
A_{_ZWADR}(cons(X, XS), cons(Y, YS)) -> MARK(Y)
A_{_ZWADR}(cons(X, XS), cons(Y, YS)) -> A_{_APP}(mark(Y), cons(mark(X), nil))
MARK(zWadr(X1, X2)) -> A_{_ZWADR}(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
A_{_APP}(cons(X, XS), YS) -> MARK(X)
MARK(app(X1, X2)) -> A_{_APP}(mark(X1), mark(X2))
A_{_APP}(nil, YS) -> MARK(YS)

Rules:

a_{_app}(nil, YS) -> mark(YS)
a_{_app}(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
a_{_app}(X1, X2) -> app(X1, X2)
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_zWadr}(nil, YS) -> nil
a_{_zWadr}(XS, nil) -> nil
a_{_zWadr}(cons(X, XS), cons(Y, YS)) -> cons(a_{_app}(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a_{_zWadr}(X1, X2) -> zWadr(X1, X2)
a_{_prefix}(L) -> cons(nil, zWadr(L, prefix(L)))
a_{_prefix}(X) -> prefix(X)
mark(app(X1, X2)) -> a_{_app}(mark(X1), mark(X2))
mark(from(X)) -> a_{_from}(mark(X))
mark(zWadr(X1, X2)) -> a_{_zWadr}(mark(X1), mark(X2))
mark(prefix(X)) -> a_{_prefix}(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:02 minutes