## Suggestion Box: Union is Commutative Proof Revisited

This is tutorial explains how to write a simple proof using the suggestion box. We will prove that union is commutative: (=_ (u. S T) (u. T S)). This is the same theorem we proved in the previous tutorial, but this tutorial proves the theorem in a slightly different way to highlight other features of the suggestion box.

### Getting Started ### Error Messages

We'll start by adding a reference to the definition of union. In Ghilbert code, this definition is found in df-un. Type "df-un" into the editor. This produces an error message: This error message explains that df-un must have three input arguments in front of it. The first two arguments must be sets and the third must be a binding variable. The proof stack can display many different types of error messages. The proof stack indicates if you have too many arguments or too few or if you're using the wrong kind of argument. For example, if you type in "S T U df-un", you will see the following error message: This error message explains that the third argument is supposed to be a binding variable, but you have used U which is a set. To fix the error use x instead of U.

• Type S T x df-un into the editor. It should now look like this: ### Copying Expressions

We now want to use the theorem orcom which says that the OR operation ∨ is commutative. The orcom theorem expects two inputs. We would like to reverse the order of two expressions: (e. x S) and (e. x T) using orcom. We could type those expressions in, but these expressions are already on the proof stack and there is a simple way to copy them. In the suggestion box, highlight this part of expression and press "Copy": This copies the expression (\/ (e. x S) (e. x T)) into the editor. That is almost, but not exactly what we want. We would like to delete the OR operation and have two seperate boolean expression. Delete the "\/" operation and type in orcom, so the editor looks like this: ### Substitution

The suggestion box now contains a new row: The bottom row of the suggestion box is for the bottom expression on the proof stack. The other row is for the second lowest expression. Whenever it is possible to directly substitute the bottom expression into the expression above it, this substitution is suggested. Both expressions share the term (\/ (e. x S) (e. x T)). The bottom expression asserts that (\/ (e. x S) (e. x T)) is logically equivalent to (\/ (e. x T) (e. x S)). By clicking the substitute button, (\/ (e. x S) (e. x T)) gets replaced by (\/ (e. x T) (e. x S)). After clicking, the proof stack looks like this: To finish off the proof, highlight the right side of the equation and click simplify. We've now shown the union is commutative.