We consider the TRS containing the following rules:
| f(g(x),g(y)) | → | f(g(x),h(y)) | (1) |
| f(h(x),g(y)) | → | f(g(x),g(y)) | (2) |
| f(g(x),h(y)) | → | f(x,y) | (3) |
| f(h(x),h(y)) | → | f(y,x) | (4) |
| f(x,y) | → | f(y,x) | (5) |
| g(x) | → | h(x) | (6) |
| h(x) | → | g(x) | (7) |
The underlying signature is as follows:
{f/2, g/1, h/1}To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| h(x) | → | g(x) | (7) |
| g(x) | → | h(x) | (6) |
| f(x,y) | → | f(y,x) | (5) |
| f(h(x),h(y)) | → | f(y,x) | (4) |
| f(g(x),h(y)) | → | f(x,y) | (3) |
All redundant rules that were added or removed can be simulated in 1 steps .
To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| h(x) | → | g(x) | (7) |
| g(x) | → | h(x) | (6) |
| f(x,y) | → | f(y,x) | (5) |
| f(h(x),h(y)) | → | f(y,x) | (4) |
| f(g(x),h(y)) | → | f(x,y) | (3) |
| f(h(y),g(x)) | → | f(x,y) | (8) |
| f(g(x),g(y)) | → | f(x,y) | (9) |
| f(h(x),g(y)) | → | f(y,x) | (10) |
| f(h(x86),g(x85)) | → | f(x85,x86) | (11) |
All redundant rules that were added or removed can be simulated in 2 steps .