We consider the TRS containing the following rules:
| +(0,y) | → | y | (1) |
| +(x,0) | → | x | (2) |
| +(s(x),y) | → | s(+(x,y)) | (3) |
| +(x,s(y)) | → | s(+(x,y)) | (4) |
| +(x,y) | → | +(y,x) | (5) |
The underlying signature is as follows:
{+/2, 0/0, s/1}To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:
| +(x,y) | → | +(y,x) | (5) |
| +(x,s(y)) | → | s(+(x,y)) | (4) |
| +(s(x),y) | → | s(+(x,y)) | (3) |
| +(x,0) | → | x | (2) |
| +(0,y) | → | y | (1) |
| +(x,y) | → | +(x,y) | (6) |
| +(x,s(y)) | → | s(+(y,x)) | (7) |
| +(s(x),y) | → | s(+(y,x)) | (8) |
All redundant rules that were added or removed can be simulated in 2 steps .