[jhanschoo]

Note that the word "coordinate" used here feels a bit disingenuous to me, because that's how one might refer to the nth property defining a mathematical object or another.

For example: The third coordinate of the rational number 1/2 is a bijection.

Coordinate here actually means: third property in the definition of a rational number in Lean. Here, this property is the statement that the denominator 2 is not zero. This is not so absurd, if we define a rational number as a tuple consisting of a natural number for the numerator (property 1) and an integer for the denominator (property 2), with the added restrictions that the denominator is not the integer zero (property 3), and that the numerator and denominator are in least terms (property 4).

But the part where the proof that the denominator is nonzero can be viewed as a bijective function, is to me indeed type-theoretic weirdness. If I'm not wrong, it's just the proof viewed as a zero-argument function. (proofs for theorems that begin with e.g. forall are functions that take arguments).

[dernett]

Lean defines a != b as a = b => False, so it seems that we have a function from proofs of a = b to proofs of False. I guess this being bijective means that there are no proofs of a = b, since there are no proofs of False, which is an equivalent way of looking at a != b.