Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
MIN(s(x), s(y)) -> MIN(x, y)
TWICE(s(x)) -> TWICE(x)
F(s(x), s(y)) -> F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) -> -'(y, min(x, y))
F(s(x), s(y)) -> MIN(x, y)
F(s(x), s(y)) -> TWICE(min(x, y))
F(s(x), s(y)) -> F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) -> -'(x, min(x, y))

Furthermore, R contains four SCCs.


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Remaining Obligation(s)
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)
       →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:

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