My first proof in Isabelle
One of the distinctive differences between academic institutions in the United States and in Great Britain is the supplementary learning outside of lectures. We have recitations in the US, which are something like extra lectures, while in the UK they have tutorials, or supervisions as they are called in Cambridge parlance. As always, they are something of a mixed bag: some supervisors are terrible, others are merely competent, and others inspire and encourage a sort of interest in the subject far beyond the outlines of the course.
Nik Sultana, our Logic and Proof supervisor, is one of these individuals. For our last supervision, on something of a whim (egged on by us, the supervisees), he suggested we attempt to prove the following logical statement in Isabelle, the proof assistant that he has been doing his research with.
I first worked out the sequent calculus proof for the statement (left as an exercise for the reader), and then I grabbed Isabelle, downloaded the manual, fired up Proof General, and began my very first proof in Isabelle.
Syntax. The first problem I had was getting a minimal theory to compile. This was because Isabelle requires you to always have an imports line, so I provided Main as an import.
I then tried proving a trivial theory, A --> A and got tripped by stating "by (impI)" instead of "by (rule impI)" (at this point, it was still not clear what 'rule' actually did).
I tried proving another theory, conj_rule, straight from the documentation, but transcribed the Unicode to ASCII wrong and ended up with a theory that didn't match the steps they did. (This was one annoying thing about reading the manual, though I understand why they did it.) Eventually I realized what was wrong, and decided to actually start the proof:
lemma "(ALL x. ~ P x --> Q x) & (EX x. ~ Q x) --> (EX x. P x)"
I first tried non-dot notation, but that failed to syntax check so I introduced dots for all bound variables.
Semantics. The proof was simple:
But that was cheating :-)
At this point, I felt pretty out-of-the-water: Isabelle uses a natural deduction system, whereas (through my studies) I had the most experience reasoning with equivalences, the sequent calculus, or the tableau calculus (not to mention I had a sequent calculus proof already in hand). As it would turn out, removing the quantifiers would look exactly like it would in normal sequent calculus, but I hadn't realized it yet.
I stumbled around, blindly applying allE, allI, exE and exI to see what they would. I hadn't realized the difference between rule, drule and erule yet, so occasionally I'd apply a rule and get a massive expansion in subgoals, and think to myself, "huh, that doesn't seem right."
Finally, reading backwards from the universals section, I realized that ==> was a little different from -->, representing a meta-implication that was treated specially by some rules, so I converted to it:
-- "Massage formula" apply (rule impI)
Once again, I tried applying the universal rules and generally didn't manage to make the formula look pretty. Then I looked more closely at the Isabelle examples and noticed they used [| P; Q |], not P & Q on the left hand side of ==>, so I found the appropriate rule to massage the formula into this form (the semicolon is the sequent calculi's colon). I then realized that there was this thing erule, although I still thought you simply applied it when the rule had an E at the end:
apply (erule conjE)
Proof. Everyone loves coding by permuting, so I permuted through the rules again. This time, exE seemed to keep the formula simple, and after a few seconds of head-scratching, would have also been the right thing to do in a sequent calculus proof. I also realized I was doing backwards proof (that is, we take our goals and break them down into subgoals), and suddenly the implication statements in the manual made a lot more sense (look at the right side, not the left!):
apply (erule exE)
This next step took a while. I was fairly easily able to apply (erule allE), which eliminated the universal on the right side of the equation, but it introduced a fresh skolem function and that didn't seem like what I wanted. I also knew that I should theoretically be able to eliminate the right-hand-side existential, but couldn't figure out what rule I should use. Trying the usual rules resulted in nonsense, though I think at this point I had figured out when to use the various variants of rule. Eventually, I reread the substituting a variable explicitly section of the manual, cargo-culted the drule_tac syntax, and it worked!
apply (drule_tac x = "x" in spec)
Experienced Isabellers will have realized that I could eliminate the right exists, but since I had concluded that this was not possible, I went on a detour. I decided to try combining some of the my assumptions to get some useful conclusions. Right now the proof state was:
!!x. [| ~ Q x; ~ P x -> Q x |] ==> (EX x. P x)
I felt I ought to be able to get P x on the left-hand side, and then apply some existential rule because of the true statement P x ==> (EX x. P x). But none of the implication rules or modus ponens applied properly. I also tried swapping over the assumption and goal using contrapositive:
!!x. [| ~ (EX x. P x); ~ P x -> Q x |] ==> Q x
Actually, I had wanted the implication on the right side. Nevertheless, the existential was on the left, so I should have been able to eliminate it... except there was a negation so the rule failed.
More reading revealed that I could use some special syntax to make the implication go on the right side:
!!x. [| ~ Q x; ~ (EX. P x) |] ==> ~ (~ P x -> Q x)
But once again, the conditional prevented me from doing my evil plan of destructing the implication and deriving a contradiction in the assumptions.
So I went back, played some more, and eventually figured out that to eliminate an existential from the goal side, you introduce it (backwards!) And I also found out that _tac could work for basically any rule:
apply (rule_tac x = "x" in exI)
At this point it was a simple propositional proof to prove, and my previous adventures with contraposative gave me an idea how to do it:
apply (erule contrapos_np) by (drule mp)
Sweet. After I finished the proof, I went back and removed the tacs and checked if Isabelle was able to unify the variables itself; it was, but the intermediate proof goals looked uglier so I added them back in. And that concluded my very first proof in Isabelle. It's not much, but it involved enough wandering that I felt quite pleased when I finished. Here's the whole thing:
theory LogicAndProof imports Main begin lemma "(ALL x. ~ P x --> Q x) & (EX x. ~ Q x) --> (EX x. P x)" -- "Massage the formula into a nicer form to apply deduction rules" apply (rule impI) apply (erule conjE) -- "Start introducing the safe quantifiers" apply (erule exE) apply (drule_tac x = "x" in spec) apply (rule_tac x =" x" in exI) apply (erule contrapos_np) by (drule mp)