Term Rewriting System R:
[f, x, y, z]
app(app(app(fold, f), x), nil) -> x
app(app(app(fold, f), x), app(app(cons, y), z)) -> app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
sum -> app(app(fold, add), 0)
prod -> app(app(fold, mul), app(s, 0))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(fold, f), x), app(app(cons, y), z)) -> APP(app(f, y), app(app(app(fold, f), x), z))
APP(app(app(fold, f), x), app(app(cons, y), z)) -> APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) -> APP(app(app(fold, f), x), z)
APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(times, app(s, x)), y) -> APP(plus, app(app(times, x), y))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(times, x)
SUM -> APP(app(fold, add), 0)
SUM -> APP(fold, add)
PROD -> APP(app(fold, mul), app(s, 0))
PROD -> APP(fold, mul)
PROD -> APP(s, 0)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(app(fold, f), x), app(app(cons, y), z)) -> APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) -> APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) -> APP(app(f, y), app(app(app(fold, f), x), z))


Rules:


app(app(app(fold, f), x), nil) -> x
app(app(app(fold, f), x), app(app(cons, y), z)) -> app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
sum -> app(app(fold, add), 0)
prod -> app(app(fold, mul), app(s, 0))


Strategy:

innermost



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