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:
| h(x) | → | e | (9) |
| g(x) | → | h(a) | (8) |
| d | → | e | (7) |
| d | → | a | (6) |
| c | → | a | (5) |
| b | → | d | (4) |
| a | → | e | (3) |
| a | → | c | (2) |
| a | → | b | (1) |
| g(x) | → | e | (10) |
| g(x) | → | h(e) | (11) |
| g(x) | → | h(c) | (12) |
| g(x) | → | h(b) | (13) |
| d | → | c | (14) |
| d | → | b | (15) |
| c | → | e | (16) |
| c | → | c | (17) |
| c | → | b | (18) |
| b | → | e | (19) |
| b | → | a | (20) |
| a | → | a | (21) |
| a | → | d | (22) |
All redundant rules that were added or removed can be simulated in 2 steps .