One can then formulate and prove a theorem that vindicates the moves your program makes. Namely, that if, after a sequence of moves, your program does terminate, then these sequence of moves corresponds to a proof, as per defined in predicate logic.

At each move, how does your program decide whether, say, to peel or to suspend? Given we agree that your program works this way modularly, then the top-level architecture of your program is not too different from a game of solitare. Given the design specification of domain-inspecificity, it is conceivable there are problems where a human peels first and other problem where a human suspends first. As such, considering all possible moves and generating a game tree, as per solitare, will avoid missing out possible winning strategies.

Let me elaborate very slightly. When we are reasoning in the propositional way that the program Mohan and I have written is designed to imitate, we often have a target proposition, an initial proposition, and some “transforming” propositions that help us get from the initial proposition to the final one. (Just to give an idea of what I mean, one could regard as a proposition that helps us to transform the statement into the statement .) This is extremely similar to what happens if we are trying to show that two quantities are equal, or that one is at most as big as another. (In fact, because and are both non-symmetric, the analogy with inequalities is possibly even stronger than the analogy with equalities.) It’s just that now instead of transforming propositions we have something more like standard replacement rules.

The analogies go quite a lot further: the operation that we call suspension above is very similar to what we do when solving an equation: we can’t instantly see what number (or other object) will work, so we give it a name like and simplify the conditions until we reach an equation with a standard solution. Something very similar happens in the propositional world if we need to specify some fairly complex object, such as a set or function, that will make a proof work.

Whether Voevodsky’s formalism would help us is less clear. In one sense (and I think this is behind Mohan’s response) it wouldn’t, since humans don’t use univalent type theory — unless they do so unconsciously, a possibility I wouldn’t want to dismiss out of hand — but in another sense it might, since it might give us a perspective on the project that would enable us to work out more quickly and efficiently how to transfer ideas from one domain (propositions) to another (terms).

Regarding your other comment… because we are interested in generating proofs in the way humans generate proofs, our approach to handling the equality of terms has been to spend a lot of time thinking very hard about how we handle equality of terms ourselves, by proving things on paper and then taking those proofs apart to try to understand what we’re doing. Methods like the one you mention may be very powerful (I’m afraid I know almost nothing about Voevodsky’s particular method), but they tend not to fit with our approach because they don’t square with how we think as humans.