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