Proof by induction over rules for mutually recursive relations

Question

Consider the (big-step) semantics of a language ($a, e$ terms, $v$ values), defined by two mutually recursive relations, $\downarrow$ and $\Downarrow$, given by a set of rule-schemata (simplified):

$\frac{a \cdot x \Downarrow e ~~ e \downarrow v}{a \cdot x \downarrow v}{\tiny PROJECT} ~~~~~~~ \frac{a \downarrow \{ \ldots x = e \ldots \}}{a \cdot x \Downarrow e}{\tiny{FLAT}}$

If I want to prove something about this semantics, I would naturally use induction over rules or judgements. However, when I arrive at rule PROJECT, I would have to make an assertion about $\Downarrow$. And to do so, I would have to return to $\downarrow$.

1. Is it legit to just merge the proofs for both relations and assume the induction hypothesis for $\Downarrow$ when dealing with $\downarrow$?

2. Is there some clean way to not merge the proofs but deal with them separately (something similar to break the recursion of functions)?

Answer 1

Yes, it's legit; no, you'll, (usually) have to "merge" the proofs.

You are trying to prove some property of an inductively defined set. You can do so using (structural) induction along the inductive order induced by your definition.

In a nutshell, you'll assume the property for all terms up to "size" (t.b.d.) $n$ (which has to apply to all rule premises) and conclude the property for terms up to size $n+1$ (the conclusions).

In your case, coming up with a good measure of size may be the hardest part, since $a⋅x⇓e$ and $a⋅x↓v$ are of the same "size" if you count operators. What size measure will work is impossible to say without seeing the whole calculus, though. Of course, for structural induction you don't strictly need such a measure; you can just follow the inductive order (see here for examples). As long as your semantics is well-defined, that should work.