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