Term Rewriting System R:
[x, y]
app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(fact, 0) -> APP(s, 0)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
APP(fact, app(s, x)) -> APP(*, app(s, x))
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(p, app(s, x))
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(app(*, app(s, x)), y) -> APP(+, app(app(*, x), y))
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(*, x)
APP(app(+, x), app(s, y)) -> APP(s, app(app(+, x), y))
APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Furthermore, R contains two SCCs.

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

The following remains to be proven:
• Dependency Pair:

APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

• Dependency Pairs:

APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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

The following remains to be proven:
• Dependency Pair:

APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

• Dependency Pairs:

APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))

Rules:

app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Strategy:

innermost

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