Term Rewriting System R:
[X, Y, Z, X1, X2]
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfirst(X1, X2)) -> FIRST(X1, X2)
ACTIVATE(nfrom(X)) -> FROM(X)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

ACTIVATE(nfirst(X1, X2)) -> FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(nfirst(X1, X2)) -> FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The following rules can be oriented:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{nil, 0}
{nfirst, first} > {activate, from} > nfrom
{nfirst, first} > {activate, from} > cons > {FIRST, ACTIVATE}
{nfirst, first} > {activate, from} > s

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
FIRST(x1, x2) -> FIRST(x1, x2)
nfirst(x1, x2) -> nfirst(x1, x2)
s(x1) -> s(x1)
cons(x1, x2) -> cons(x1, x2)
first(x1, x2) -> first(x1, x2)
activate(x1) -> activate(x1)
from(x1) -> from(x1)
nfrom(x1) -> nfrom(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Dependency Graph

Dependency Pair:

Rules:

first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes