We consider the TRS containing the following rules:
| a | → | b | (1) |
| a | → | c | (2) |
| a | → | e | (3) |
| b | → | d | (4) |
| c | → | a | (5) |
| d | → | a | (6) |
| d | → | e | (7) |
| g(x) | → | h(a) | (8) |
| h(x) | → | e | (9) |
The underlying signature is as follows:
{a/0, b/0, c/0, e/0, d/0, g/1, h/1}To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| a | → | e | (3) |
| b | → | d | (4) |
| c | → | a | (5) |
| d | → | e | (7) |
| g(x) | → | h(a) | (8) |
| h(x) | → | e | (9) |
All redundant rules that were added or removed can be simulated in 4 steps .
Confluence is proven using the following terminating critical-pair-closing-system R:
There are no rules.
There are no rules in the TRS. Hence, it is terminating.