Why does MatchQ[a, r_ /; Head[r] != Plus] evaluate to False?

Question

I'm trying to understand. Why is it that:

MatchQ[a, r_ /; Head[r] != Plus]

Evaluates to:

False

? For me, I would think that, because:

Head[a]

Evaluates to:

Symbol

Where a has no value or expression assigned to it, then this:

MatchQ[a, r_ /; Head[r] != Plus]

Should evaluate to True. Could someone point me in the right direction to understanding this better? Thanks!

Answer 1

The problem is this:

If a is a symbol then its Head is Symbol, so Head[a] != Plus reduces to Symbol != Plus. Unequal (!=) is supposed to be a mathematical test for inequality. In this case, it just cannot decide whether Symbol != Plus should evaluate to True or to False, since both sides are Symbols. Here an example why this is undecidable with the current amount of information:

With[{Symbol = 1, Plus = 1}, Symbol != Plus]
With[{Symbol = 1, Plus = 0}, Symbol != Plus]

False

True

So the expression Symbol != Plus stays unevaluated. (This is the best strategy since later definitions could make it decidable.)

Because the second argument of Condition (/;) does not evaluate to True, the pattern does not match.

Lesson to learn: For testing for structual inequality, use UnsameQ (=!=):

MatchQ[a, r_ /; Head[r] =!= Plus]

True

Of course, the same applies, mutatis mutandis, to Equal and SameQ.