### Simply adding lambda won't make your rewrites diverge

I've been working on trying to extend a result from my thesis, having to do with strong normalization of a certain higher-order rewrite system, but its proof is very complicated, so I've been working on simplifying it.

Along the way I found a paper by Mitsuhiro Okada from 1989 which proves strong normalization for the combination of STLC with any first-order term-rewriting system. But this paper is hard to read because it has strange notations and concepts and says things like, "If a part s[*/x1, ..., */xn, */y] of t belongs to the cap then the part s[*/x1, ..., */xn, */y] also belongs to the cap." Say what?

I knew I could prove the same result using the Tait-Girard method (the same basic approach Okada takes) in a more straightforward way—at least to me. So I did, and here it is for future reference: "Simply adding lambda won't make your rewrites diverge."