proofs

It took us some time to get to write an actual proof.

There are many kinds of things to prove: in the form of con/disjunction or implication. If we want to prove something in con/disjunction, then we have to break it in different cases, to discuss all the situations to complete the proof. The most direct and clear way is to prove something in implication. First we need to assume the prerequisites ( like the set of our objects ). If it is something like "for all", then we have to " assume" it because this is the precondition. Every time we assume something we need to remember to indent on next line. We also need to assume the antecedent of the implication and then we try to lead it one way down to the consequent to complete the proof. The most important process is to connect the antecedent with the consequent. Finally we conclude all the preconditions we've assumed and the whole proof is done.

The most interesting ( also the most difficult ) part of a proof, as to me, is picking some thing for "there exists". At the beginning of the proof we'll usually construct a line like "pick......" and Danny suggests us to leave it blank and go on with the proof! This sound crazy but actually it's a very helpful strategy because at the very beginning of a proof one can barely know what kind of value to pick for a variable. Thus we leave the line blank and during the proving process we can figure out what kind of value to pick and fill in the blank.