Term Rewriting System R:
[x, y, f, g]
app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(fix, f), x) -> APP(f, app(fix, f))

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Non Termination`

Dependency Pairs:

APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))

Rules:

app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Strategy:

innermost

Found an infinite P-chain over R:
P =

APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))

R =

app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

s = APP(app(fix, fix), app(fix, app(fix, fix)))
evaluates to t =APP(app(fix, fix), app(fix, app(fix, fix)))

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:01 minutes