Term Rewriting System R:
[x, y, h, i, u, v]
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, x)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(or, true), y) -> true
app(app(or, false), y) -> y
app(app(union, empty), h) -> h
app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) -> false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(ifreach1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(ifreach2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(ifreach2, true), x), y), app(app(app(edge, u), v), i)), h) -> true
app(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(eq, app(s, x)), app(s, y)) -> APP(eq, x)
APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(app(edge, x), y), app(app(union, i), h))
APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(union, i), h)
APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(union, i)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(ifreach1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(ifreach1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(ifreach1, app(app(eq, x), u)), x), y)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(ifreach1, app(app(eq, x), u)), x)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(ifreach1, app(app(eq, x), u))
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(eq, x), u)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(eq, x)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(ifreach2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(ifreach2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(ifreach2, app(app(eq, y), v)), x), y)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(ifreach2, app(app(eq, y), v)), x)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(ifreach2, app(app(eq, y), v))
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(eq, y), v)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(eq, y)
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, x), y), i)
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, x), y)
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(reach, x)
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(edge, u), v), h)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(or, app(app(app(app(reach, x), y), i), h))
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), h)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, x), y), i)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, x), y)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(reach, x)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, v), y), app(app(union, i), h)), empty)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, v), y), app(app(union, i), h))
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, v), y)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(reach, v)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(union, i), h)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(union, i)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(union, i), h)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, v), y)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, v), y), app(app(union, i), h))
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, v), y), app(app(union, i), h)), empty)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, x), y)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, x), y), i)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), h)
APP(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(reach, x), y)
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(reach, x), y), i)
APP(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(eq, y), v)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(ifreach2, app(app(eq, y), v)), x)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(ifreach2, app(app(eq, y), v)), x), y)
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(ifreach2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(ifreach2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(eq, x), u)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(ifreach1, app(app(eq, x), u)), x)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(ifreach1, app(app(eq, x), u)), x), y)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(ifreach1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> APP(app(app(app(app(ifreach1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(union, app(app(app(edge, x), y), i)), h) -> APP(app(union, i), h)
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)


Rules:


app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, x)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(or, true), y) -> true
app(app(or, false), y) -> y
app(app(union, empty), h) -> h
app(app(union, app(app(app(edge, x), y), i)), h) -> app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) -> false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(ifreach1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(ifreach1, true), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(app(ifreach2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(ifreach1, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(ifreach2, true), x), y), app(app(app(edge, u), v), i)), h) -> true
app(app(app(app(app(ifreach2, false), x), y), app(app(app(edge, u), v), i)), h) -> app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:04 minutes