Term Rewriting System R:
[X, Y, X1, X2]
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
alength(nil) -> 0
alength(cons(X, Y)) -> s(alength1(Y))
alength(X) -> length(X)
alength1(X) -> alength(X)
alength1(X) -> length1(X)
mark(from(X)) -> afrom(mark(X))
mark(length(X)) -> alength(X)
mark(length1(X)) -> alength1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0) -> 0

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AFROM(X) -> MARK(X)
ALENGTH(cons(X, Y)) -> ALENGTH1(Y)
ALENGTH1(X) -> ALENGTH(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(from(X)) -> MARK(X)
MARK(length(X)) -> ALENGTH(X)
MARK(length1(X)) -> ALENGTH1(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pairs:

ALENGTH1(X) -> ALENGTH(X)
ALENGTH(cons(X, Y)) -> ALENGTH1(Y)

Rules:

afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
alength(nil) -> 0
alength(cons(X, Y)) -> s(alength1(Y))
alength(X) -> length(X)
alength1(X) -> alength(X)
alength1(X) -> length1(X)
mark(from(X)) -> afrom(mark(X))
mark(length(X)) -> alength(X)
mark(length1(X)) -> alength1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0) -> 0

Strategy:

innermost

The following dependency pair can be strictly oriented:

ALENGTH(cons(X, Y)) -> ALENGTH1(Y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
ALENGTH1(x1) -> x1
ALENGTH(x1) -> x1
cons(x1, x2) -> cons(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

ALENGTH1(X) -> ALENGTH(X)

Rules:

afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
alength(nil) -> 0
alength(cons(X, Y)) -> s(alength1(Y))
alength(X) -> length(X)
alength1(X) -> alength(X)
alength1(X) -> length1(X)
mark(from(X)) -> afrom(mark(X))
mark(length(X)) -> alength(X)
mark(length1(X)) -> alength1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0) -> 0

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

The following remains to be proven:
Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
AFROM(X) -> MARK(X)

Rules:

afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
alength(nil) -> 0
alength(cons(X, Y)) -> s(alength1(Y))
alength(X) -> length(X)
alength1(X) -> alength(X)
alength1(X) -> length1(X)
mark(from(X)) -> afrom(mark(X))
mark(length(X)) -> alength(X)
mark(length1(X)) -> alength1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0) -> 0

Strategy:

innermost

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