Expansion of \ifnum creates a weird \relax token

Question

There was I, procrastinating over some TeX code while I should have been writing my dissertation. The code was working more or less fine (better than the writing, at least), until a mysterious \relax popped up from nowhere!

At first I though it was something stupid from my end, but when I reduced the code to a bare minimum I realised I had no idea what was going on. I reduced the code to use basically only primitives, so it shouldn't be any coding problem. Here's the guilty code:

\def\useIInnn#1#2#3{#2}
\def\useIInn#1#2{#2}
\detokenize\expandafter{\ifnum0=0\expandafter\useIInnn\fi\useIInn{1}{\BOOM}}
\bye

and its output is (surprisingly, for me):

\relax \fi {1}{\BOOM }

Where did that \relax come from?

---

I realise that the test wouldn't result in 0=0 because \ifnum would continue expanding tokens (as far as I understand, it would make 0=01 and result false; I corrected this in the code :). However the \relax remains a mystery for me.

Answer 1

Let's see what happens. The tokens \ifnum0=0 are not a full test, because by rule TeX expands tokens until finding an unexpandable token that cannot be interpreted as a digit; if this token is a space, it will be swallowed. However, TeX will never go past the matching \else or \fi when expanding tokens in the process of determining a conditional test.

The token following 0 is \expandafter, which expands the \fi: oh, this means that the conditional has to be evaluated! In such cases, that is, whenever a conditional text is unfinished at the time \else or \fi appear and have to be expanded, TeX inserts a special \relax token, called frozen \relax.

You get a frozen \relax also in cases such as \if x\fi; two in the case of \if\fi.

This is module 379 in tex.web.