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