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