Termination Proof using AProVE

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

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(cons(X, XS), YS) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
PREFIX(L) -> NIL
PREFIX(L) -> PREFIX(L)
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_nil}) -> NIL
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳Remaining

Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)

Rules:

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

The following dependency pairs can be strictly oriented:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, n_{_nil}))
ACTIVATE(n_{_zWadr}(X1, X2)) -> ZWADR(X1, X2)
ACTIVATE(n_{_app}(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{cons, ZWADR, n_{_zWadr}, n_{_nil}} > {APP, ACTIVATE}

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
APP(x₁, x₂) -> APP(x₁, x₂)
n_{_app}(x₁, x₂) -> n_{_app}(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
n_{_zWadr}(x₁, x₂) -> n_{_zWadr}(x₁, x₂)
ZWADR(x₁, x₂) -> ZWADR(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Remaining

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

PREFIX(L) -> PREFIX(L)

Rules:

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

Termination of R could not be shown.
Duration:
0:00 minutes