Term Rewriting System R:
[y, x, z]
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(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> 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(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

Innermost Termination of R to be shown. 



   R
     ↳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(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(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(minsort, app(app(cons, x), y)) -> APP(cons, app(app(min, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(min, x)
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(del, app(app(min, x), y))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(if, app(app(le, x), y))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(le, x)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(min, y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(if, app(app(eq, x), y))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(eq, x)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), 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(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> 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(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes