## The Problem

Consider the java program max:

int max(int a, int b) {
if (a > b) return a;
return b;
}


and its mutant

int max(int a, int b) {
if (a >= b) return a;
return b;
}


These compile to respective PEGs (phi (> (var a) (var b)) (var a) (var b)) and (phi (>= (var a) (var b)) (var a) (var b)). I’d like for Cornelius to prove that these are equivalent programs.

While it looks like it would be trivial to prove equivalence, it’s actually kind of tricky to do so soundly with in an EGraph. The main difficulty is that I want my EGraph to reason about (var a) locally in the then branches of the mutant’s phi node and discover that (== (var a) (var b)) whenever (&& (>= (var a) (var b)) (! (> (var a) (var b))).

I can get the mutant into a more workable form by applying a few rewrite rules:

(phi (|| (> (var a) (var b))
(== (var a) (var b)))
(var a)
(var b))


and at a high level, all I need to show is that both versions of max act the same when (== (var a) (var b)). To do this I want to replace the (var a) in the then branch of the mutant with the (var b). Explicitly, I’d rewrite the || node into something like

(phi (> (var a) (var b))
(var a)                      ;; || short circuits
(phi (== (var a) (var b))    ;; when (! (> (var a) (var b )))
(var a)                  ;; we can replace (var a) with (var b) here!
(var b)))


If I can rewrite the then branch of (phi (== (var a) (var b)) (var a) (var b)) to (var b), then I get the form (phi (== (var a) (var b)) (var b) (var b)). This always evaluates to (var b) so I can replace the entire phi node with (var b). Substituting this back in to the above program, we transform the mutant into (phi (> (var a) (var b)) (var a) (var b)), which is our original program.

The question is: how can I soundly rewrite the above then branch? This involves local reasoning: reasoning about (var a) differently inside of the then branch of the phi node than anywhere else in the program. This turns out to be difficult to do in an EGraph because it’s easy to accidentally identify things globally.

## Question

What techniques can I use to employ local reasoning in an EGraph? Solving this is going to be perhaps the crucial task for Cornelius to work. Many other things are ‘mere’ engineering, but this problem seems like it is going to be the crux of the matter.

Possible solutions include:

1. Equality Refinement: Equality saturation allows us to use conditions such as (== a b) in a phi node’s condition to replace instances of a with b inside the then branch.
2. Condition Replacement: Similar to equality saturation, a node (phi c thn els) replaces each instance of c with true in thn and false in els.
3. Spawning New EGraphs: If we can spawn a new EGraph for a place here certain identifications are made and then extract the resulting identifications, this could be profitable.