Termination Proof using AProVE

Term Rewriting System R:
[y, x]
app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(le, x)
APP(app(minus, x), app(s, y)) -> APP(pred, app(app(minus, x), y))
APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(if_gcd, app(app(le, y), x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(le, y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(gcd, app(app(minus, x), y))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(minus, x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(gcd, app(app(minus, y), x))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(minus, y)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:
innermost
Dependency Pairs:
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:
innermost
Dependency Pairs:
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:06 minutes