Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FROM(X) -> FROM(s(X))
AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Instantiation Transformation

       →DP Problem 2

         ↳Remaining

Dependency Pair:

FROM(X) -> FROM(s(X))

Rules:

from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Strategy:

innermost

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

FROM(X) -> FROM(s(X))

one new Dependency Pair is created:

FROM(s(X'')) -> FROM(s(s(X'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Inst

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(X'')) -> FROM(s(s(X'')))

Rules:

from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Strategy:
innermost
Dependency Pair:
AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)

Rules:

from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Inst

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
FROM(s(X'')) -> FROM(s(s(X'')))

Rules:

from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Strategy:
innermost
Dependency Pair:
AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)

Rules:

from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Strategy:
innermost

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