We consider the TRS containing the following rules:
| a | → | c | (1) |
| b | → | c | (2) |
| f(a,b) | → | d | (3) |
| f(x,c) | → | f(c,c) | (4) |
| f(c,x) | → | f(c,c) | (5) |
| d | → | f(a,c) | (6) |
| d | → | f(c,b) | (7) |
The underlying signature is as follows:
{a/0, c/0, b/0, f/2, d/0}To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| a | → | c | (1) |
| b | → | c | (2) |
| f(x,c) | → | f(c,c) | (4) |
| f(c,x) | → | f(c,c) | (5) |
| d | → | f(c,b) | (7) |
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.