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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(nfst(X1, X2)) -> FST(X1, X2)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nlen(X)) -> LEN(X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(nlen(X)) -> LEN(X)
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfst(X1, X2)) -> FST(X1, X2)
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
activate(nfst(X1, X2)) -> fst(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nlen(X)) -> len(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(nlen(X)) -> LEN(X)
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfst(X1, X2)) -> FST(X1, X2)
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
cons > {LEN, s, ACTIVATE}
{FST, nfst}

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
LEN(x1) -> LEN(x1)
nlen(x1) -> nlen(x1)
nfst(x1, x2) -> nfst(x1, x2)
FST(x1, x2) -> FST(x1, x2)
s(x1) -> s(x1)
cons(x1, x2) -> cons(x1, x2)

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

Dependency Pair:

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
activate(nfst(X1, X2)) -> fst(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nlen(X)) -> len(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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