Term Rewriting System R:
[X, XS]
zeros -> cons(0, zeros)
tail(cons(X, XS)) -> XS
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
tail(cons(X, XS)) -> XS
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(tail(x1)) | = 1 + x1 |
| POL(zeros) | = 0 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳RRRPolo
→TRS2
↳OC
→TRS3
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
ZEROS -> ZEROS
Furthermore, R contains one SCC.
R
↳RRRPolo
→TRS2
↳OC
→TRS3
↳DPs
...
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
ZEROS -> ZEROS
Rule:
zeros -> cons(0, zeros)
Strategy:
innermost
As we are in the innermost case, we can delete all 1 non-usable-rules.
R
↳RRRPolo
→TRS2
↳OC
→TRS3
↳DPs
...
→DP Problem 2
↳Non Termination
Dependency Pair:
ZEROS -> ZEROS
Rule:
none
Strategy:
innermost
Found an infinite P-chain over R:
P =
ZEROS -> ZEROS
R = none
s = ZEROS
evaluates to t =ZEROS
Thus, s starts an infinite chain.
Non-Termination of R could be shown.
Duration:
0:05 minutes