Term Rewriting System R:
[YS, X, XS, X1, X2, Y, L]
app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(X) -> X

Innermost Termination of R to be shown.

R
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, nnil))
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(napp(X1, X2)) -> APP(X1, X2)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nnil) -> NIL

Furthermore, R contains two SCCs.

R
DPs
→DP Problem 1
Remaining Obligation(s)
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
• 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, nnil))
ACTIVATE(napp(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(X) -> X

Strategy:

innermost

• Dependency Pair:

PREFIX(L) -> PREFIX(L)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(X) -> X

Strategy:

innermost

R
DPs
→DP Problem 1
Remaining Obligation(s)
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
• 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, nnil))
ACTIVATE(napp(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(X) -> X

Strategy:

innermost

• Dependency Pair:

PREFIX(L) -> PREFIX(L)

Rules:

app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil