[DanWaterworth]

> Dependent types doesn't explain Coq's requirement for pure, total functions.

That's where you are wrong. Partial functions introduce the bottom value and then you can 'prove' arbitrary incorrect things.

[Locke1689]

I'm not sure what you're saying. I was saying that simply being dependently typed doesn't necessitate pure total functions. Obviously, these features can be considered either highly suggested or necessary for ITPs. There are a number of ways you can get around totality with dependent types.